While Braunvieh's are good cattle and more well known of the two, I believe Aubracs are better muscling, thicker, easier calving, more thrifty, have a more moderate frame score, and hardier. Braunvieh's are typically larger (frame) and mature later than Aubracs. Both milk well. Aubracs are typically more efficient at feed conversion (to milk and muscle). The Braunvieh breed association appears to be more organized and more people breed them.
While I see that Braunvieh's have scored well in various carcass tests, I have no experience with actually eating fullblood Braunvieh beef. I can tell you is we're pre-sold out of our production for the next two years already -- fullblood Aubrac beef is good eatin'.
I can't speak about their temperament, but I can say that Aubrac breeders have (and continue to) cull any attitude problem. As a result, the temperament of the Aubracs are, generally, quite good. We can pet our bull and he won't even look up from grazing.
Like comparing any two breeds, the best cattle in either breed are going to be exceptional cattle, period. The worst cattle in either breed should have been slaughtered last week. So, no matter which breed you go with, look for exceptional animals to use in your breeding program.
Good luck, and LMK if you have any questions about Aubracs. I'm happy to help as I might.
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A couple of years ago while bringing home some older bred fullblood Aubrac cows, I had a great time arguing with a good ol' boy at a gas station in some small town in Iowa about the breed of cattle in my trailer. I told him I had Aubracs. He told me I had short, fat Braunvieh. I told him I didn't. He said I did. We went back and forth for a while like that, and then when I went in to pay, I heard him tell everyone else filling up that I didn't know what sort of cattle I was hauling and they were short, fat, Braunvieh. It gave me a good chuckle.