The B77 is a more modern version, dating from about 1979 until sometime in the 1990s. It has logic controls and won't stretch the tape if you switch from fast wind to play or something. It can also be remotely controlled, I'm not sure if the A77 can do that.
Some people say that the A77 sounds a bit better, I can't really judge. I have a B77HS mk2, but it's faulty unfortunately and I haven't figured out how to fix it. Fast wind was having problems on mine, and the pinch roller solenoid was underpowered as well. It might be as simple as replacing some capacitors, but I'm not sure. I needed to have a mixdown deck so I got an A807 instead.
What you might want to look out for is the PR99, especially the mk2 or mk3 versions. These are the professional version of the B77, and the mk2 and mk3 have a digital counter (the mk1 and the B77 are mechanical), plus return to zero functionality.
There were some of these going on that BBC auction, in fact.