With the growing legitimacy of games as a storytelling medium, there's been a lot of talk about what games are capable of doing to tell stories that other mediums simply can't. A novel can give you inner dialog far better than a play or movie, while a movie lets you tell expansive, sweeping stories with far too many locations to ever be used in a play. Etc. etc., the list goes on.
Games can most easily be compared to movies. You have as many locations as you need to tell to tell the story, the story mostly occurs through outwardly visible events (an excess of internal monologue can bog the story down), and it lends itself fairly easily to spectacle. The question comes when you ask what facets of storytelling games lend themselves to more than anything else. Games aren't movies, but the easiest way to approach the question is to see how they differ importantly from them.
Games are longer – Though nowhere near all of the content is storytelling, games can easily tell stories that are far too long to be contained in a movie. This is not to say that they need to; Portal doesn't contain enough material to make any kind of a film, but no one can question the quality of the writing, despite the very simple story. (This actually necessitates the question of “writing” versus “storytelling”, but that's for another time)
Games get you attached to the main character much faster – This appeals to common sense, but I'd be interested to see some psychological study to back it up. When you control a character's movements, it creates the illusion of controlling the character's fate. Even if the gameplay is completely linear, you still control the fate of the character in terms of whether you pass the next challenge the game brings to bear. We identify with well written characters because they resemble us in some way. It doesn't matter that none of us have ever been charged with carrying a ring of ultimate evil; we can all look at Frodo, and remember times when we had a huge burden unwillingly thrust upon us. We want him to rise to the occasion because we want to think that, in the same situation, we would be able to rise to the occasion as well.
When a character does what we tell them to do, that bond forms that much faster. Curiously enough, this phenomenon manifests even if the player has minimal control over the fate of the character. The line waffles between wanting to complete the next challenge, and wanting Gitaroo Man to pass the next challenge, despite the fact that your “control” over Gitaroo Man consists of turning the analog stick and hammering arbitrary buttons to make him dodge clumsily. Nonetheless, I am more eager to pass the game's challenges because they also represent the struggles of this fictional character, as opposed to abstract hoops to jump through.
Games open up the possibility of non-linear story structure – Not since the choose your own adventure book has there been as much buzz as there is now about non-linear story structure. I'm not really going to open up the huge can of how best to do that, but suffice it to say that the holy grail of narrative design in games would be a game in which the player has full reign of the world around him, and the world reacts to anything the player wants to do, even going so far as to put dramatic emphasis on the events the player cares about, and tending to skip over the ones the player has spent less time engaging in.
No one has quite nailed this one on the head yet. It seems that no one has even really broken the code of how to approach a story from this angle. There's been a lot of interesting discussion on the topic, but I'm not here to push my opinion on what constitutes a non-linear storyline. The question that pops into my mind is “Does a game's story have to be non-linear?” A truly non-linear storytelling model would be revolutionary, but it wouldn't necessarily push older models out of the spotlight. Many gamers today grew up playing entirely linear Japanese RPGs, some of which (though by no means most) are still revered for the quality of the story. Likewise, if a non-linear story hit the market, it wouldn't attract very much attention if the story wasn't good to begin with. If you have an infinite number of options for potential fates of a character you couldn't care less about, well... that doesn't make people start caring.
All that being said, it seems that all this discussion of a good non-linear storytelling model isn't going to do a ton of good until the concept of writing in games has achieved a greater level of legitimacy. It seems a bit hackneyed to say it, but I'm not sure pushing the envelope will accomplish much until the current medium has been mastered. Until it becomes general knowledge that an excellent, emotionally engaging story sets a game apart, conceptual, high level thought about the nature of non-linear storytelling will be largely wasted on the mindset of a great deal of game studios.