So is the short story full of spoilers for the game? Is the game full of spoilers for the Story? Which one should I enjoy first?
I'm going to second this question, because I'd probably get around to reading the story first (almost done with game 2 in the Deponia trilogy)
I read the short story some time ago (though I'm getting ready to re-read it again tonight! Thanks, GOG!) and from what I recall there are some major differences between the story and the game -- mostly character development, minor plot details and narration perspective --, but I can safely say you're free to either play the game before reading the story or the other way around without one getting in the way of -- or spoiling -- the other (well, there's some spoiling to be had, but that's inevitable, for obvious reasons).
For... "chronological" reasons, I guess?, maybe read the story before playing the game, but it actually doesn't matter all that much, they're quite standalone experiences on their own.