Arguably, the most unique capability of games when compared to other media is their interactive nature. Because of the chaotic nature of player demands, games to this point have typically provided a very linear experience, hidden or otherwise, to ensure that the player experiences what the developers want them to experience.
Even games that are open-world, such as Fallout 3 or GTA IV still have a very directed story that the player chooses to progress through over the course of the game. But, stories are being told every minute of every day, we just need to look at it with different eyes.
A recent example of ceding authorial control to the player is the Left 4 Dead franchise. Valve provided a set of unique locations, interesting theme, characters in the game, giving players all of the tools they need to write a story. The AI director ensures that the pacing and drama is provided uniquely with each play of the game.
The characters provide contextual responses based on the players current performance in the game. The key difference is, however, that the story comes not from anything delivered directly by the developers, but from the experiences by the player (or players) during each play through.
Some might say that Left 4 Dead doesn't provide true authorial control - the player can't deviate from the provided content, they cannot change the story or the characters interactions beyond what has been provided by the developer.
These are all established limitations for any game, and is ignoring the strength of games when compared to other media: the authorial power of games doesn't end with the content provided by the developers, that is where it begins. True authorial power comes from player interaction - how the player progresses through the game is where the real story for games is derived.
Looking at my last example, Left 4 Dead (but certainly NOT limited to L4D, one could use any of the Civilization games, or The Sims series, or even the Battlefield series) merely provides a context and set of mechanics for the players to deal with. The story is derived from how the player deals with the challenges established by the relatively (or explicitly) dynamic nature of these games.
The story is further randomized by the choice of playing solo or cooperatively, and can be even more unique if played in an adversarial match. The stories that are created are unique with nearly every level played, and only change further based on the player's decisions in playing the game.
Of course, some might say that the stories generated by these sorts of games aren't really stories at all. But aren't they? They have a protagonist, or protagonists, and antagonists. The have an exposition, a climax and denouement. The story changes based on who is participating - in that, I would pose that the stories created and told by these sorts of games are closer to the oral traditions of the ancient storytellers. More importantly, these stories are based around the one thing that makes games unique: interactivity.