I have never heard you say this before. Would you explain both:
1) What story is in relation to gameplay and atmosphere.
2) Why atmosphere is the counterpart to mechanics.
3) Why videogame criticism is fucked as a result of the confusion.
Honestly curious. I like this train of thought.
I think this actually taps into a fundamental theory about videogames, stuff I may have a hard time putting into words, but I'll try to keep it as short and simple so as not to stress myself too much.
First I need to highlight the problem with "story" in that it is a very poorly defined term in game discussions (so is gameplay, for that matter) and no one bothers to think about what they are saying versus what someone else is saying. This is why virtually every thread on the topic looks like a complete mess. People say "story is bad" and end up meaning a bunch of different things: characterization, dialogue, plot, tone, cutscene direction, setting, logic in and out of plot scenes, concern towards so-called "mature" topics, sensitivity toward so-called "taboos", etc. It is no surprise then that threads on the subject boil down with the thoughtless, automatic understanding that "videogame stories are bad" (because any of the listed qualities) where people argue for exceptions (because any of the listed qualities). (It gets less stupid the more the subject is narrowed down.)
I'll relate it the other way around. Atmosphere could be described as the feeling or mood of the world where interactivity occurs, in a way it is what makes a world that world. It suffers from a bit of ignorance where people think only rainy, foggy, or spooky scenes have it, but I'm talking about it at a much higher level. All games have some minimum of an atmosphere, even the most barren Atari-looking crap (that's what it means to have a very thin layer of atmosphere, not Call of Duty, not even close). Theoretically you'd put the entire audiovisual aspect under here, what the interactivity and feedback of a game looks and feels like. That would include sound design, music, art style, visual fidelity, everything listed above, etc. It may be seen as the "skin" of the mechanics, but it can also considered irreplaceable to a game's feedback, thus the emotion the player feels. I should take a moment to say emotion is another one of those poorly understood things about videogame criticism where people think in very limited terms; It's a laughable scenario like atmosphere (how silly the term "emo" is just about describes it well). The best games (the games you think are the best before having even read or considered this post) are the most emotional ones.
Building on that, the relation between story (whatever that may actually contain) and atmosphere is one between a means and an ends, respectively. All these elements you'd find parallels of in films exist to make objects come alive and become impressionable, to turn an exchange of numbers into an engaging conflict, set up interactivity, depict results, etc. Thinking like this is a paradigm shift: a story doesn't exist to be a "good story" (I encourage thinking about what is the goal of a story in any sense), it exists to be apart of a videogame. A "good videogame story" is not necessarily a "good film story" or "good novel story", it's whatever the videogame needs it to be, because the videogame ultimately isn't trying to tell the story (not even the ones filled to the brim with cutscenes), it's trying to make you feel something through its interactivity and feedback (which is draped in atmosphere, which may utilize an assortment of things that can be considered "story"). One of the most damaging (or decadent) ideas to come out of this carriage before the horse approach to story and atmosphere is that "story-telling" is considered an end goal of games (and I could go a step further and say this applies to more mediums than games, but that's taking this too far than I care to). It leads to all sorts of absurd conclusions, like putting movie-like before life-like. To put it simply, people are too concerned with a game's "story" (no doubt in posturing hindsight overly concerned with "tropes", "messages", and whatnot) and not with the phenomenon of it, even if that works directly against it being a neat, sensible little device of story-telling. Note that I'm only criticizing how people see games, not the games themselves. Those "story-centric" games are not necessarily unfavorable even under this new understanding, they should just be understood differently.
I'd like to turn my attention towards a particular set of denizens of NeoGAF, people in this thread, who are also infected with this toxic form of thinking. Often you'll hear "I only care about the gameplay" for games like Bayonetta, Dark Souls, etc. I see this and think the pseudo-intellectuals have successfully stolen a piece of the medium from enthusiasts. Outside perhaps a few irregular individuals, anyone who says this doesn't really understand why they like the games they do. The atmosphere for the two examples and countless others are fundamental to what makes those games what they are, fundamental to the way they were designed and meant to be enjoyed. Everything from music to attack animations were painstakingly chosen and implemented, as it is with all the best games. Indeed, it is more likely the case the game began with an atmosphere and the mechanics followed. Perhaps people think this way because they are in the mindset "gameplay vs. story", where story is so easy to call bad (and I'd challenge even that, even for Bayonetta). This is plainly shown to be ridiculous, because enthusiasts will retreat from story discussion threads to their enthusiast threads, like this one, and freely talk about how sexy Third Strike (oh yeah, this thread is about fighting games) with a degree of passion no less then why they like any one mechanic of the game. They'll savor animated GIFs (you know, where there is no "gameplay") of cool scenes (QTE scenes, even).
Anyway, the opposite of mechanics is generally considered aesthetics, although I subscribe to the idea that the two cannot be fully separate (they seem to become more deeply integrated the more I think about it). I have hard time nailing down a distinct difference between aesthetics and atmosphere to honest, but I do feel there is a difference, it is just which is which in relation to mechanics can flip on me (one's higher/lower, one is more isolated from mechanics) and the difference may just be a matter of how you look at something. Better left unsaid.
EDIT: I want to emphasize I think there is a great incompatibility between "game stories" and the stories of other mediums. What ends up happening is that people messily "rip out" the story of a game, internalize it as a movie, then subject it to very superficial or agenda-centric film criticism (look at the displeasure people have with modern day film criticism to see what I mean). Not only would I say this is poor form, but as explained, I'd say it is entirely backwards and counterproductive in what it is trying to do. There's also the matter I didn't touch on of "maturity" and "juvenile power fantasies", a whole can of worms in itself. I'd say atmosphere-first (or emotion, stimuli first) has a way making that kind of thinking less relevant (which is a good thing).