Yes, they amount to the same thing. If you say 'I was brought up' you are also putting your parents (and whoever brought you up) into the picture, but the essential information is exactly the same. By the way, I presume when you say 'pretty small town' you mean 'relatively small town', as opposed to a small town that is also attractive?

Brought up by your parents, was taught by them. grew up does not mean that you were taught. I see difference in it. You can grow with your parents, but be brought up by your uncle and aunt