So if the question is: What is better, AA between Europe and US compared to BA:

In my opinion AA has one of the worst F products. The seats are pretty good and about on par with BA and others, but everything else sucks. Service is almost always bad, sometimes even rude. Food is more like business class food on bigger plates, lounges are mostly a joke and so on.
BA in my opinion is just average. Seats are small (lots of other airlines have the suites type now which offer lots of space - 8 or 9 seats where BA has 14). Food is better, Service is a bit hit and miss, but still lightyears above AA.

If you need to pay for the ticket look at Virgin. Overall experience in Business (Upper) Class similar to BA, better than AA and cheaper.