I think the major quality points have already been mentioned. Just a warning, I have flown mostly in J or F on either BA or AA in the last 10 years so I can't really give you a specific comparison on the economy experiences of the two airlines.

In general, compared to AA, the BA crews are more warm, friendly and consistent. I have had great services over the years on AA but also shockingly bad ones like how is this person still employed bad! For example, once I had a flight attendant in first who consistently mixed up what I ordered.. flat water instead of sparking, white wine instead of champagne, cold breakfast instead of the hot breakfast, etc. If I didn't know it was an AA flight, I would have thought she had some underlying neurological condition.

Since you are flying in economy, the overall product I believe isn't all that different in quality from what's offered on AA or Delta. Although BA used to offer London Pride which is one of my favorite beers.