I'm not sure that there's a correct answer to that one, but for my two cents: the one thing that other mediums, those of "fine art", do well that video games have been severely lacking in is emulating reality. Many gamers are adults and claim that games can be made for "just adults", but most games (even those made for "just adults") still retain a huge array of fantasy bells and whistles aimed at kids. While most gamers don't mind because they're just used to it or might not even notice it, to the outside world, that's a big part of why it's still largely perceived as juvenile.
There are whole genres of painting, cinema, poetry, literature, etc. dedicated to just making something beautiful or interesting out of commonly occurring things from real life, and video games have not (with much success) found an engaging way to do this yet. Just about every video game I've ever played has had some kind of fantasy, sci-fi, horror or adventure element. There are some obvious exceptions, like sports games, historical RTS's, sims - but they don't put much focus on making the game's world reflect any real life experience (unless you're a pro-athlete, warlord, city planner or God).
Even the games that ostensibly take place in a more realistic world (think COD/MOH military shooters or Rockstar games) rely so heavily on film tropes that they really cannot be accepted as a reflection of the real world, just a reflection of a reflection. Even though much of cinema also indulges, there are many films and filmmakers who excel at making the world feel real and appeal to adults who aren't as interested in over-the-top. While there are many things that games can do just as well as (or even better than) cinema, this is not one of them.
For example, Hitman has a lot of excellent voyeuristic passages, and used to have a very subdued tone that made you feel like it could possibly be taking place in the same universe as a Martin Scorsese, Michael Mann, or even a Coen Brothers film... but then there's all this retarded pseudo-sci-fi shit about him being a supersecret superhuman clone or whatever.... and those nuns...
Spider-Man 2 (I know, superhero, but still) was an awesome game that gave you free-roam exploration of Manhattan, but it didn't quite make me feel like I was breathing New York air the way Taxi Driver, Annie Hall, or even Ghostbusters did.
A game like Heavy Rain is kind of a step in the direction I'm talking about, but even still, it deals with especially dark subject matter that a normal person can't really relate to. Obviously, there has to be some kind of out-of-the-ordinary event that drives the plot, otherwise it would just be real life and that would be boring, but I don't think game developers and writers have really tested themselves enough to see how realistic they can go - or how interesting they can make real things. Maybe a game where you work in a hospital and have to deal with patients in a high-stress emergency environment (one that isn't a cartoon, thanks), or a straight-forward detective game (L.A. Noire was so close, but had that overtly cinematic vibe and really linear crime solving). Even committing crimes can be done more realistically - the new release of Thief comes to mind because you could set that game in the present day real world, or at the very least not make your protagonist look like a ridiculous ninja-pirate, and have it be just as engaging.
Bottom line is: there isn't any real tangible thing that separates "fine art" from any other art, it's just in how it's perceived (and by whom). I don't necessarily think that an art-form must comply with these standards to gain status as a "fine art", but it definitely helps people perceive it that way - particularly academia, who generally has more leverage to make those calls than critics or general audiences.
tl;dr - It's the stupid clothes your favorite game heroes wear.