There is no major difference between these two. But I have never heard 'born and brought up' used before. I'd recommend using one of these two: 'Born and bred' OR 'Born and raised'

Contrary to what Justin says, "Born and bred" is a set phrase that has often been used in English when speaking about people. But that said, it is not heard very commonly these days. "Born and raised" is heard much more often, and is also something of a set phrase. "Born and brought up" is not really a set phrase, but they both work as independent verbs, and in your example, I think they sound fine together.

"Breeding" is used in American English when one selects certain animals to have sex to produce a desired genetic result. Very common with dogs, cats, horses, and cattle. Idk if the other person who answered is from somewhere else. We would never say "born and bred" here. "Born and raised" is good.

"Born and bred" and "born and raised" are both commonly used. But I'm not sure I've ever heard anyone say "born and brought up."