In general, they mean the same and are interchangeable. You can raise or bring up (or rear) a child, children, or a family.
We do differentiate the terms in some standard phrases:
Let's get married and raise a family.
I was born and raised in California.
Otherwise, they are synonyms, as well as the verb 'to rear', as in to rear children, to rear a family. 'To rear' may not be used as often as it once was, and it sounds kind of old fashioned.
You can also bring up or raise a question. But you don't rear a question. And in class you raise your hand to ask a question, you do not bring up your hand.