I've personally never understood fandom on fandom bullying. When it boils down to it, all fandoms are the same.
As far as the whole Bronies = Furries thing, yes, you can be a furry if you're a brony. Furries share a common love of all things anthropomorphic (Giving human characteristics to non-human creatures/things. i.e. tiny horses that can talk). However, the inverse is not true, you can't be a brony just by being a furry. Think of Furry as the meta-fandom for all things anthropomorphic and brony as another branch in the tree.
The whole Anti-Brony movement that's supposed to be a big thing in furry is blown way out of proportion. My ponysuit gets so many hugs, pictures, and bro-hoofs at fur cons. I've so far only met one furry who is so furiously anti-brony he storms out of the room and away from me when I show up in ponysuit.