It's not just simple interactivity, otherwise something like Jack White's new interactive music video (which is smurfing awesome, btw) would be a video game.

I think one of the most important components is the repercussion factor. The player's choices must either shrink the list of available choices (via loss of life, injuries--includes Mario shrinking et al) or expand it.

The number of choices in a movie is static. The actions themselves are static.

But choices in games are context-sensitive. Different outcomes result from exactly the same actions. Take the input: Mario jumps.

Under a breakable block: Mario Jumps --> block breaks
Under a ? box: Mario Jumps --> mushroom appears
In front of a gap: Mario Jumps --> Mario eaten by the abyss
Under a pipe: Mario Jumps --> Mario is sucked into the underworld
Under a jumpy turtle: Mario Jumps --> Mario shrinkage
etc