After sitting through preferences, the distinction is really one which social contract is used/preferred. The idea that one or the other is somehow more mentally challenging is a bit laughable - RPG rule sets are written in good faith, often by disparate authors, with varying production values, and subject to a staggering range of GM and player interpretation without any real authority to recourse to. Being able to break them in one manner or another really isn’t an intellectual distinction for either CaW or CaS. It may be an illusion of personal competence to pretend one or the other is somehow more OOC challenging.

As to CaS/CaW as a social contract, that is where the difference lies.

A CaS social contract implies that the GM will give the players a “fair” range of encounters (be they fights or fast taking your fantasy meth dealer). The players can expect that so long as they behave in a semi-reasonable fashion, they will not suddenly get hammered with a “GOTCHA!” about not explicitly packing their healing potion, or being wiped away with a hellfire from an unanswerable drone 30k feet in orbit with no chance to do anything about it. In turn, the GM expects the players to keep the plot moving without infinite paranoia and preparation, and that they try to solve the issues they encounter in good faith. That doesn’t mean the answer to every guard is a stabbing - bribing, sneaking, seducing, mind controlling etc. are all fine so long as players are not deliberately trying to break the encounter. And this can honestly be a very low-stress way to play a game about doing cool things in cool places. The players know that if they are genre savvy their fun isn’t going to be an illusion, and the GM knows that he may have to plan a world, but at least it won’t be one that players are trying to break. It can also be seen as stifling by some, who feel that they are being inadvertently railroaded, or that social conventions at the table are guiding their actions more than they would like.

CaW is a different form of social contract. The players have no expectation that anything about it will be fair. “ROCKS FALL” is a possible outcome from seeming to follow the narrative arc. The GM plays the opposition to the limits of their resources...and maybe a little more. The players in turn are expected to find every trick they can, IC and meta, to win. Even if this means going knowingly against the theme of an encounter or generic table conventions.
This can, on one hand, be highly liberating. You are free to legitimately tricks those players into an ambush that they as real world people didn’t see coming, and then machine gun their helpless characters! You are free to solve the GMs elaborate Kobold Dungeon by flooding it with poison gas and sealing the cave mouth! It can also be obnoxious for some people - it can slow down a table as people try to cover every possibility, it can frustrate players when they thought an obvious social cue like “talk to the elaborately described NPC in the corner” should not immediately lead to “shot in the head”, and frustrate GMs when for real or imagined possible IC reasons a plot grinds to a halt or is entirely discarded.

To use a non-combat example, let’s look at a quest/mission/shadow run/whatever.

In a CaS contract, there is an understanding that “taking the quest” is going to be the generally done thing. You’re in front of the duke, he wants you to solve the goblin problem, and you know that the GM probably has tonight’s events planned around solving the goblin problem. Oh, you may be betrayed, ambushed, lied to, reneged on, drakes, run out of town and so forth as a result of all this, but these are narrative components that come with a bit of plot insurance from the GM. Since it’s part of the contract that you’ll take the quest, his side of it is that the act of accepting the job isn’t going to be fundamentally game ending.

In a CaW contract, you’re still in front of the duke who wants you to solve his goblin problem. You have every right to tell him to pound sand; you don’t like the feel of it, you think he’s treacherous, he won’t pay enough, the goblins sound extra tough, whatever. The GM has to deal with that. However, if you take the quest and the Duke betrays you, you’re getting shot twice in the back of the head as you come out of the goblin cave. Roll a new character and do better client research next time. Caveat emptor. The player has to deal with that.

Done right, either contract can bring a whole lot of fun to a table. Done wrong, and well the potential weaknesses of each are pretty glaring.