Aside from seeing starting pitchers attempt to hit 2-3 times a game in the NL, I don't see a lot of difference in the quality of games or strategy between the leagues.
After the starting pitcher leaves, so the NL manager has to go to his bench for pinch-hitters (temporary DH's, in other words) and occasionally employ the double-switch (something any 3rd grader could master). Oh, and occasionally you see a rally get killed in the early/middle innings of a NL game, because there are two outs and either the #8 hitter or the pitcher are due up. That's about it.
Some say that there are more fastballs in the NL, and maybe some NL teams have more of a fetish for stocking the bench with speedy utility types, but from my seat in the stands (or at home) those details don't affect my enjoyment of the game all that much.