I picked combat.
Doing it for the story is very much a mixed bag. I've not played in a game with a story that satisfied me. I very strongly dislike "I was a farmer yesterday, now I have a sword and I need to go kill some rats in a cellar, so one day I might face a dragon", and yet at the same time, I'm not interested in "Saving the world every day that ends in Y."
I get a fun idea for a character, I yearn to play those characters with arching stories and such, but at the table I just end up saying what my actions are and being a little more quiet than necessary.
As a DM, I try to be very consistent and obsess about the story, but players don't always buy that, and so on. I ran a game where everyone were gypsies, and the "story" wasn't as interesting as the players just coming up with con artist jobs and being goofy. I want my PCs to be gripped by the story, to really care about the outcome, but that's a lot of work, on everyone's part, and it's very hard to pull off.
And at the end of the day, I'm frustrated, I want to blow off some steam, and I just want to roll dice, kill some monsters, and be cool. Or I want to play those awesome, awesome monsters.
So I'll be honest: Combat, unless something is interesting.
I also think that D&D as a whole just doesn't facilitate stories, doesn't facilitate characters as much. In every edition, 80% of the rules focuses on Combat, combat, combat. So, it's just not the system for that. If I wanted to run a game based on character or story, I'd use Spirit of the Century. Or Exalted. Or Dresden Files RPG (a story based game I hunger for). Or a myrid of others before D&D.