Why do cats lick people?

When a cat licks you, it can serve several purposes. Grooming, affection (which grooming is a part of), tasting, and leaving their scent on you.

Especially kittens use licking to taste new things. Even if their sense of taste isn't as developed as that of humans, they still behave like human children and learn about new things by tasting them.

If you meet other cats while you're away, your cat will smell them on you -- even if you didn't really cuddle the other cat, since a cat's sense of smell is about ten times as acute as ours -- and will "mark" you as HER (or his) human, so the other cats don't think you're a stray and try to adopt you ;p If you did pet the other cats, your cat will cover their scent with her (or his) own, and at the same time learn as much as possible about the other cat.

Cats groom their kittens to care for them, to keep them healthy and to show them they are loved. A cat will do the same to you, of course, since she likes you and wants to show you that she cares about you. =)

I don't know that much about dogs and why they lick people though ^^ So I can't say if they have different reasons or not (well, slightly different, I guess).