Sadly, the current trend is towards games as design. The ever-increasing power of real-time 3D rendering allows more and more realistic spectacles to appear on the screen. Therefore, this is really the wrong time for this question to be brought up. A glance at the latest offerings for any system will reveal games to be about as artistic as the photography in a Sears catalog; that is, just enough to get by.
But in times past, and in a few places in the present, games are indeed artistic. Never "high art," of course, whatever that is... but the oddball 3D cutout style of Parappa the Rapper is art. The hand-drawn sprites of the latest 2D fighting games are, occasionally, art. The character design for some games is definitely art -- look at the original drawings for Metal Gear Solid or King of Fighters 2001 -- and that art is at least somewhat carried over into the game's final appearance. And, of course, some games have music that goes beyond simply setting the mood, and plays a part in telling the story. (The example I keep repeating every time this topic comes up: In Final Fantasy 6, each character has a leitmotif. At the very end of the game, two of the characters have a brief interaction, not enough by itself to imply that they've fallen in love, but during this interaction, their two themes play together, and it turns out that they harmonize beautifully. The resolution of that subplot is told perfectly and clearly through music.)
And if you believe that games cannot inspire other works of art, then either you've never played an inspirational game (the Final Fantasy series is an apt, if hackneyed, example) or you don't have the mindset to be inspired by these games (which I don't intend as an insult). A quick look at the fan art and fan fiction sections of the GIA, for instance, will show a good deal of artwork and writing inspired by video games... much of it awkward and amateur, of course, but some of it surprisingly good.
Of course, the single greatest example of game as art would be the Mortal Kombat series... ooh, blood! Look, mommy! Blood! With that kind of crap standing as spokesman for video games, is it any wonder that the topic of games as art is controversial?
<edit>And yes, in some games you can see the designer/author very clearly... sometimes to an extent you wish you hadn't. It takes a very influential (aka egomaniacal) designer to be able to leave his personal mark on a game, but every time you interact with a precocious and beautiful scientific genius in Metal Gear Solid 1 or 2, for instance, you're talking to Hideo Kojima's repressed ideal dream woman. Other subtle examples abound. And of course, whenever you encounter an interestingly designed character or plotline or music, you're seeing an expression of the creator's experience and personal sensibilities. If, on the other hand, you're talking about art as autobiography, then no, games aren't like that, but neither is all art.</edit>
[This message has been edited by Perfect Thunder (edited 12-10-2001).]