It's like what Phearia said. There really is no real difference if you raised it or caught it outside of the achievements. What I've been doing is just replacing pets, that can be replaced, as I go along. However, I favorite (star) the pets, that I know can only start at level 1. Those are the ones I would raise for the achievements. Right now, I've got the Curious Wolvar Pup at level 20. When I reach the top level, I'll start looking for the rare versions of the pets I have. The rare versions definitely have a boost in stats all around but no special abilities compared to the same pet of lesser quality.