The downside to being undead would of course be gradually falling apart. Vampires of course have this problem beat as the conveniently heal whatever injury they have.
The humble zombie has a much harder road to travel. Part of the horror behind the zombie monster is their horrific rotting bodies and our own fears of death and putrification. Compared to zombies vampires have it easy these days. They get to be good looking, rich, immortal, and overly emotional Bellas just throw themselves at them.
Ain’t nobody lining up to date a zombie.
Until recently that is. The humble zombie is slowly making inroads into the “Teen paranormal romance” section of the bookstore. You have books like “Warm Bodies” and “Breathers” treating zombie more sympathetically than merely walking targets for our gun fantasies. Of course in order to do make zombies more acceptable to teenage girls, you’ve got to clean them up a bit. Febreeze helps. And of course you have to do something about that pesky rotting apart thing.
I’ve read a couple stories recently where they get around that problem by simply making a rule that if their zombie characters “Eat” they don’t rot and they regenerate. If they skip some meals… then they start getting gross.
The tradeoff in these stories is that the sexier authors try to make zombies the less horrific they become. You simply cannot have your cake and eat it too.