I'm a big JD buy so I obviously will go with JD. What sets Drew apart for me (because I think of both as pretty clutch players) is his ability to get on base, coupled with his great fielding. He's got one of the best right field arms in the league, strong and accurate, gets good jumps on everything, plays awkward liners well, and plays smart. Despite the myth that he's a wimp and doesn't care, he IS willing to dive for balls, and runs all grounders out. JD is what he is as a batter. I can see why his approach frustrates people, but the fact is he does what he has to do, "not commit outs" (as Theo would say).
It's also interesting, when you think about it, how people associate JD being soft with him missing games, when Mr. Dirt Dog Trot missed a ton of games with injuries.