Well, there's the weight. A low Bb bari a lot lighter than a low A one.
I find that low Bb baris tend to be more 'mellow' in tone - and as such I've always felt that they're a better bet for soloing...at least within the mainstream genre.
I would have previously put this down the the simple difference between vintage baris and modern ones, but I find it rings true with the few modern low Bb baris out there these days.
That said, I have a Conn-Selmer Avant 180 on the bench right now, and that seems to have quite a warm tone. But that was on blowing it when it came in, complete with leaks. It'll be fixed later...and I'll give it another go.
A low A bari will get you more gigs these days.
If something gets wedged down the bottom of a low A bari it's much harder to reach in and get it out.
The majority of low A baritones have dreadful mechanisms for actuating the low A via the thumb key only. Yamaha's mech is sublime.