I think my favourite thing about him is that he is never afraid to be himself, no matter how different he is to any normal dog. He does all kinds of crazy things that dogs don't normally do, and sometimes other characters pick up on this and even seem irritated by it (e.g. Frieda when she keeps trying to encourage Snoopy to chase rabbits like a normal dog). He never gives into the norm, though.
I think there's a good lesson to be learned there. :)