The whole dynamics of the AL game would change, and teams would be directly affected by it. The White Sox couldn't have both Thome and Konerko, the Blue Jays couldn't have Overbay and Big Hurt, the Tigers would have to use Sheff exclusively in the outfield, Boston would have to use Ortiz at first (where he's terrible), the Indians would have to use Hafner at first... I could go on and on and on. Having a DH gives you one extra hitter in the lineup, and that changes everything. In the NL, the 9th hitter is an easy out (unless its D-Train or Jason Marquis

), but in the American League, the 9th hitter is used almost as an extra leadoff man- just looking at tonights Sunday Night Baseball game, the Yanks are using Cabrera and the Bo Sox are using Crisp at the 9 spot. It just gives you a huge advantage having a DH because your lineup becomes that much stronger.
Also, having a DH allows injured/fatigued players to get a day off essentially but still while batting in the lineup. Today the Rangers used Young at DH, so he was still in the lineup but wasn't playing the field. This is a huge advantage that AL teams have, as they are able to rest their players better and allow players with a nagging injury (see Johnny Damon) to not miss time.
Either have it in both leagues or don't have it at all. I like to joke about how boring NL baseball is but not having a DH is a more "pure" way of playing the game. The DH almost seems cheap to me, but I would like to either see it in both leagues or neither leagues.
I have a question too- does anyone know if the DH can be applied to a position besides the catcher? Like if you have a great hitting pitcher, could you DH the catcher and just have the catcher field but not hit?
"Oh, that Lankford and McGee, the trio of 'em. They're a one-man wrecking crew."
-Mike Shannon