There are three areas to cover as far as I see it for this question. What the games themselves are about, similarities they share, and then how the games differ from each other.
You'll have to forgive how verbose I'm going to be here. I figure this is a good opportunity to make as comprehensive an answer to this as possible. Not only for your benefit but so that I can link others to this as this question comes up again and again. Especially when the game releases I feel it will get a lot of use. So strap in for quite a read.
Chapter 1: What Are These Games About?
Shadow of the Demon Lord
SotDL is a horror fantasy game largely inspired by Dungeons & Dragons and Warhammer Fantasy Roleplay. Both games the author, Robert J. Schwalb, worked on a good deal. The basic premise is that it's set during the final days of Urth, a world that has fallen under the threat from the titular Demon Lord. The Demon Lord is an unimaginably powerful cosmic entity that exists in the spaces between realities, called the Void, and is essentially entropy incarnate and whose sole desire is the end of all creation. Fortunately Urth is protected by a magical barrier that keeps between it and the Void, unfortunately this barrier is not perfect and the cracks in it allow the Demon Lord's influence, and its demon offspring, to seep through and corrupt the land as its shadow spreads.
However, all hope is not yet lost. While things are dire the end of days is not a certainty and exceptional people from all backgrounds will rise to defend the world. Whether they come from prestigious academies, bloody battlefields, or simple farms and fields the coming apocalypse can thrust them into greatness. These champions need not be good people either as the end to all things will make allies of even the most bitter foes. Righteous paladins and vile necromancers might stand side-by-side to fight back against the spreading evil, whether that involves tracking down potent relics, exploring ancient faerie ruins, journeying into Hell to bargain with the Devil, or battling demons and the cultists who summon them.
It's a horror fantasy through and through, it runs the entire gamut of horror too and gives you options for basically anything you could want. It's got body, cosmic, and sexual horror, it's got gore, revulsion, and shock, it does lovecraftian nightmares, undead of every kind, and debased humans of every stripe, and mostly everything in between. Some of that stuff isn't going to be for everyone but it never forces any one style on a group and with just its core book it's mostly about scary monsters with a bit of the other stuff around the edges. Other supplements might get more specific and hone in on a certain theme but if that's not your thing you'll just skip those things.
Shadow of the Weird Wizard
SotWW is a dark fantasy game largely inspired by the Dungeons & Dragons setting Greyhawk. A setting the author, Robert J. Schwalb, has a good deal of love for. The basic premise is that it's set after the disappearance of the Weird Wizard. He is an enigmatic character of godlike power that used his magical prowess to claim an entire continent for his own. Using the land as a test bed for myriad experiments from creating creatures of every size and shape, raising a great city and populating it with his automata slaves, even reshaping the land so mountains reach the stars and rivers of stone cut through the plains, and other such feats. However his disappearance caused, was because of, or merely coincided with the greatest civil war Erth has ever seen. While the Weird Wizard may no longer be around his shadow still looms large and the conflict has sent refugees fleeing from the Old Country in their thousands to the Borderlands, a place that had once sheltered under the protection of the Weird Wizard. His disappearance has also acted as an invitation for the more adventurous among them to venture into the lands he left behind, full of wondrous treasures, dangerous monsters, impossible landscapes, and possibly a new home for those uprooted.
So in the aftermath of a great war, a massive influx of peoples to a land that can't really support, and the disappearance of such a major figure things are in a fairly unstable place. In times of great instability there are always great heroes that arise to help rebuild and that is where the PCs come in. Your characters will go on to find great fortunes, build a legend for themselves, and leave a mark on history. But the great destinies that lie ahead of you start with humble beginnings. While you will soon be on your way to slaying legendary creatures, finding vast riches, wielding powerful artifacts, and fulfilling noble quests you have to start somewhere. Being forced to settle new lands would be hard enough without your village stumbling upon an ancient idol that infects your friends with a thirst for violence, or the threat of beastmen dragging off your family, or children getting lost in woods that never seem to end. How your characters choose to solve these problems will be their first steps to greater adventures.
Tonally this game is "grey fantasy". While there is plenty of darkness to be found it doesn't revel in it and it doesn't assume you'll be using it. And on the other hand the player characters are assumed to be generally noble or at least working towards noble ends. They're characters who can, and will, produce positive outcomes and make Erth a better, safer, place to live in. But it's also not just darkness to contend with as there is a lot of space for the wonderous, magical, or otherwise strange that may or may not be a threat. So the game is evoking a lot of classical heroic fantasy tropes of going on great quests, with a backdrop of widespread turmoil and crisis, in a setting full of strangeness and mystery.
Chapter 2: Where Are They The Same
Both games are fairly streamlined, medium-crunch, d20 games. You only need d20's and d6's to play. d20 is for your checks and tests, the d6's for damage and boons/banes. A boon is any beneficial circumstance, and a bane is any hindering circumstance. In combat boons are typically derived from talents and buffs, and banes from afflictions, debuffs, or attack options. Boons/banes are d6's you roll alongside your d20. They cancel each other out 1-for-1, so you never roll a mix of boons and banes. For a boon you add the highest value among all your boons, and for a bane you subtract it. So if you have 3 boons, and they roll 2, 4, 4 you simply add 4 to your d20. These are the major modifier, and the only modifier outside of a stat mod that's equal to Stat - 10. And speaking of stats there are 4 to worry about here, Strength, Agility, Intellect, and Will.
All rolls are resolved in one of two ways. "Attack" rolls are d20 +/- mod +/- boons/banes vs a target attribute or characteristic. For weapon attacks this will usually be Defence, but spells will target attributes directly, and so can other abilities. Then you have "challenge" rolls which are always vs 10. Attack rolls are for attacks or actions that would otherwise be opposed, challenge rolls are for literally everything else. Skills, resisting deleterious effects, etc. That is the core system in a nutshell.
Each game eschews traditional initiative systems in favour of a streamlined system in which you can choose to act before your enemies but only take a Move OR an Action, or afterwards and take both a Move AND an Action. Speaking of, there are also only three types of things to do in a turn. Moves, Actions, and then a third action type that works like bonus actions and reactions from D&D 5e all rolled into one. However the games do offer a lot of baseline actions you can take from the get go. Weapon attacks also have lots of prebaked options to use as well in order to provide martial characters more flexibility.
Another thing the games don't do is traditional skill systems. Instead you get Professions. A profession is simply a job (Hunter, Beggar, Minister, Scholar of X, etc) and when something happens that your profession applies to you get an amount of boons (Boons are d6, if rolling multiples you just add the highest. Banes are the same in reverse and they cancel out 1 for 1) as the GM sees fit, or an auto-success. They don't give direct combat benefits, being largely narrative and RP focused, but can be used in combat indirectly. In a skill system you might have +5 Perception, in SotDL it would be City Watch or Hunter. Both of those professions require being alert and watchful but in totally different settings and scenarios along with other unique skills and experiences for each profession. A member of the City Watch would have knowledge of where to find gambling dens or other criminal activity, they'd have a certain amount of authority in which to manipulate peasants and the like with, they'd know other towns and cities would organised their own watches, where as the Hunter wouldn't know any of that yet they'd both have a keen eye.
Another major component of the system is the way characters are constructed. These games lack a traditional class system in favour of a tiered system. You get a Novice "Path" at level 1 which is a broad archetype (Magician, Priest, Rogue, Warrior). At 3 you get an Expert which is more specialized and flavourful than a Novice are, and these are more like traditional classes in theme and mechanics (Wizard, Berserker, Assassin, Paladin, etc). Then at 7 you get a Master Path which is a specialization (Infiltrator, Pyromancer, Exorcist, Weapon Master, Diplomat, etc). As you progress in tier your options expand too, in the core book you've got 4 Novices, 16 Experts, and 64 Masters. There are no restrictions and Warrior > Fighter > Blade is just as valid as Magician > Assassin > Brute. Notably a Path doesn't just give you levels until you get a new Path. Your Novice Path, for example, doesn't just give you benefits at levels 1 and 2, but when you hit level 5 it'll also provide you something there. So your full leveling experience is made up of these Paths intertwining.
The system is built ground up to allow any combination you want, and for you to still end up competent regardless. You really have to go out of your way to make a bad PC, and that's intentional because a large part of this game is choosing your Paths based on your story and experiences in game. Both games have a big focus on narrative and you're given as much freedom as you could possibly want in how you represent all of that. There are synergistic choices, but hyper-specialized builds have their own drawbacks.
The games also start you without a level or Path and so all you have for additional mechanics is derived from your Ancestry and Professions. You also determine your starting gear based on rolling on a variety of tables. Now is probably the best time to mention it but weapons in both games are fairly simple affairs of Xd6 damage, the amount of hands it takes, and a couple of properties to alter exactly how it works.
Magic is also handled a little differently to a lot of standard fantasy games. There are no class-based spell lists. Instead these games divide spells into "Traditions” which are smaller spell lists that are unified by theme and general mechanics. Things like Celestial, Chaos, Enchantment, Necromancy, Primal, or Technomancy. As each has its own mechanical purview you simply can't have all the answers, if you didn't choose a Traditions that grants you the ability to fly, you're not going to get to fly unless you buy an incantation/inscription (scroll) or they give it to you. So everything has its own strengths and things it can and can't do. Not only is this great for the batman wizard issue, it also lets PCs who chose exactly the same Paths play very differently, it's also another avenue for customisation and creating flavourful PCs.
Magic items are also rarer than a lot of fantasy games. Both these games have three major sorts. You've got potions and minor things of that nature you can buy. Then there are objects that have minor to good effects but are typically random both in form and effect. Of these you're only going to have a few each. Then you've got much more potent items that are both unique and storied, things that entire campaigns could revolve around, and of these you might not even all get one.
They share some setting elements too, most of the overarching cosmological elements are the same. A countless number of realities all closed off from an endless expanse of nothingness in which the Demon Lord and its host dwells destroying worlds. Probably because they're all in the same overarching setting. Although my current theory is cyclical realties. Then you've got immortal faeries that come in all shapes and sizes that are largely alien and amoral and for various reasons don't mingle with mortals. To contrast the singular but unaging life of an immortal mortals age, die, but reincarnate after being ferried through some underworld or other. Faeries have some sort of war with trolls who are almost universally vile warmongering creatures. Demons spill in from the Void sometimes and cause all sorts of problems. Gods are largely absent but are very real. etc. etc. etc. Tonally these things are handled very differently but there is a good deal of overlap between the two in specific elements.
Chapter 3: How Are These Games Different?
Usually, and as you've already seen, it's easier to talk about this as things that have changed from SotDL to SotWW because the nature of design is iterative. However that always involves caveats that SotWW isn't just SotDL 2e. But given you've got little familiarity with both of these games it's much simpler to do direct comparisons. For the record though, in case you were under the impression that SotWW was SotDL 2e in some fashion: It's a new game in a modified version of the system doing its own thing and while there are things the author thinks are improvements to SotDL but most of it is designed for separate goals in mind. With that in mind I'm just going to go through Chapter 2's points and talk about how the games might differ in the specifics. Should you like more detail about why things are different feel free to ask.
Power Level
This is one of the biggest mechanical differences of the two. SotWW features PCs that have a much more impressive set of abilities in comparison to SotDL's. Everything from Ancestries, Paths, and Spells all pack more of a punch than their equivalent level in SotDL. SotDL doesn't have PCs that feel particularly weak by any means but SotWW PCs are a clear step above. This does naturally apply to enemies too. Things you might think are terrifying in SotDL are often monstrously so in SotWW. Naturally this also means that PCs in SotWW are typically hardier than those found in SotDL. While death isn't off the table in SotWW it's harder to get
Rolls
SotWW includes both attack and challenge rolls, with new terminology, but also a third type called luck. Luck rolls are challenge rolls without a stat mod. Largely used to determine things like when and effects might end, when you regain access to a talent after you've used it, it might be used in addition to a standard roll and let you pick either option, and general checks just to see if you got lucky.
Initiative
Both games have a slightly different turn order but do use the same style of turns. In SotDL it goes PC fast turns > enemy fast turns > PC slow turns > enemy slow turns > end of round. In SotWW it's PC fast turns > enemy slow turns > PC slow turns > end of round. In SotWW it also takes the use of your reaction to take a fast turn.
Actions
Both games have a lot of actions going on but the major difference here is how they handle the aforementioned attack options. In SotDL when you make a weapon attack you can take a bane on it to do something extra, pushing enemies away, tripping them up, etc. In SotWW this ties into a new mechanic called "combat tokens". These are essentially the way martial Paths scale their damage. The thing you can do the most with them is just spend them to hit harder. But you can also spend an amount of them to do the same sorts of attack options. Hitting for less damage but not hitting less often, unlike in SotDL. SotWW does in general have a more robust pool of standard actions too.
Professions
These work very similarly in both games but in SotWW each category of Profession does also provide a unique talent to use. However you only get one of them, unlike SotDL's 2 + extras from Paths, but Paths themselves now serve similar purposes.
Weapons
SotWW's weapon system is as described above but it differs from SotDL in that the weapons have more complexity in terms of the amount of properties, and the effects of them. It's nothing major but weapons in SotWW do have more going on. Lots of effects that trigger on "critical hits''. They also interact with the various ways you can spend combat tokens. One weapon might be cheaper to knock back enemies with than another which might be quicker and more able to attack multiple foes. Stuff along those lines.
Paths
The tiering is the same here but the spread of levels is different. In SotDL Novice Paths grant benefits at 1, 2, 5, and 8. Expert Paths at 3, 6, and 9. Master Paths at 7, and 10. In SotWW Novice Paths are 1, 2, and 5. Expert Paths are 3, 4, 6, and 9. Master Paths are 7, 8, and 10. The idea behind this change is that SotWW's higher power levels necessitate some more room for mechanics on Expert and Master Paths to fully sell their themes, and giving 2 levels back to back in each Path helps the PC feel more like that role sooner.
Ancestries
SotDL handles Ancestries in a very standard way with 1 real exception. Ancestries dictate starting stats, have a chunk of unique mechanics associated with them, and all that good stuff. The major exception here is that in SotDL your Ancestry is the source of your level 4 benefits, which you might have guessed as it was missing from the above. In SotWW Ancestries are now divorced from stats and those are chosen independently and what an Ancestry grants is just its unique mechanics. Ancestries do not grant level 4 benefits in SotWW either.
Path Mixing
So while there are no restrictions in either, SotDL and SotWW have different mechanics in other places that have altered the efficacy and ease in which you can combine Paths to produce competent characters. SotDL's biggest problem here is the way magic scaling works. There is a stat called Power that dictates the rank of spells you can learn, and Novice Paths grant you the most spells. Which combine to mean that if you don't take a magic Novice Path than a later magic Path is rarely going to be more than a trap option. SotWW effectively solves this issue but I'll get into that next.
Magic
Both of these games use Traditions in the same manner but the spells themselves are handled differently. In SotDL spells have Ranks from 0-5 and if you take nothing but magic Paths you'll get 1 Power every odd level. This Power dictates the Rank of spell you can learn and the amount of times you can cast each spell you know based on its Rank. SotWW splits its spells into 4 tiers to correspond with Paths, and a lower Starting tier. There is no Power and instead the tier of Path you're at dictates the tier of spells you can learn. So where as in SotDL if you took a Master Path as your sole magic Path you'd be stuck with some very weak spells for the level, in SotWW you'll get as high of a tier as everyone else does. Additionally, spells in SotWW have fixed castings that don't increase as you advance.
Magic Items
Not much to say here other than SotWW just includes more magic items as purchasable things.
Other Things
SotDL uses grid based movements with range bands for weapon and spell distances as well as various shapes and sizes for AoE effects. SotWW uses zone based movement and uses Zones for range measurements and AoE effects will be whole zones or pick targets within a zone. SotDL includes basic rules for Insanity and Corruption in its core book, and expanded rules in supplements, which are fitting for a horror game and as such no analogue exists in SotWW. There are also a billion little other things here and there too but it's impossible and pointless to cover all of them in that sort of detail.
I think that basically covers it.
