It might sound like a fighting game-elitism thing or something, but I can't describe it any more than saying it's different. The timing on pulling off a huge combo in a fighting game is so far removed from the computationally simple matter of aiming and shooting in an FPS that online play completely loses the complexity. For example, check out this guide to a character in Marvel vs. Capcom 3 and take a peek at some of the videos if you can. Looking at the string of inputs, and realizing that every single one of those is measured in frames and fractions of frames, it should be apparent why dropping a single packet of information could be vital. Zero has some of the more complex combos in MvC3 and, back when I was interested in the game more, he was considered a high-tier character in offline play, but executing his advanced combos was nearly impossible online, even if you could get it perfectly in local play every time. You have to vastly adjust your timing and at best only get part of what makes his combos so powerful. A character with less complex inputs automatically gets raised in tiers, often far more than they ever could have been, while complex and timing-intensive characters frequently get shafted. Competitive fighting game play online simply is not the same as locally, and local competition used to be available in arcades.
That is incredibly complex, and must take quite a while to get down and be able to pull off with any reasonable amount of complexity. I understand what you're saying with local competition being, essentially, better than online competition for fighting games. We can agree on the matter of it being different. At a certain point in FPS games, you reach a level where everybody can aim and shoot with precise accuracy. At a high level it becomes a game of anticipating your opponent, much as in the realm of fighting games, and always be a step ahead. At that same point winning and losing also becomes a function of teamwork, and having an executable plan that could be altered at any second.
Yes, that's exactly it. FPS, RTS, Fighting, and other genres all require different skillsets to be competitively viable. Although execution (the ability to aim and shoot accurately, for example) is important in an FPS, execution skill is not as vital as the greater strategy of the game. In a fighter, the intensely precise execution is not just an important factor but can be the sole determining factor between a miserable noob and a good (not great) player. Every game has a different balance of strategy and execution, and it happens that fighting games have, as far as I know, the highest ceiling of execution of any genre as a result of the intense nature of good combos. There is of course a decently high strategy ceiling in well-designed fighters, too. Different genres can certainly be competitive online. In fact, all games but those with the highest execution ceiling are pretty viable competitive online games. Unfortunately, fighting games suffer massively from this, and, to me, fighters are emblematic of the once-great arcade culture in the US. You may notice that I focus on the US in my comments, because Japan and East Asia in general still have arcades and, incidentally, tend to pump out some of the best new blood. As networks get better I really look forward to new generations of players in the US scene, but the arcade scene in Japan is really amazing. Unfortunately, unique to that culture too.