I have at least 4 groups where the story is what matters.
Ah. You see, I would think the difference lies not in "what is important", but in how the game progresses, and what the player's actions (as opposed to the character's actions) are like.
You can play Settlers of Catan in such a way that the story matters, but it is neither a role-playing game nor a story game.
In a story game, the bulk of the rules are rules for manipulating the story. In D&D, the bulk of the rules are for resolving character actions - you can look at the results as a story, but you don't manipulate the story directly.