I flew AA F from JFK-LHR and then BA F (old) from LHR-ORD last week.

1. Lounge: I preferred the Concorde Room over the Flagship Lounge. Greater variety of food/beverages IMHO along with the size and general layout
2. Crews: Both were outstanding; very attentive and friendly
3. On-board: I preferred the BA OF to AA. I love flying in the nose of a 747 over the 777... Something about the AA seat was extremely uncomfortable to me and left my back in spasm after we landed in London. YMMV of course.
4. Food: AA has stepped up it's game and the food was good, wines were ok. BA was also good so it was perhaps a tie here.
5. IFE: A tie I'd say. Small screens and what seemed to be limited selection. I had my iPad along so it really didn't matter much to me...but I also understand that NF on BA has a big screen and better selection. (NOTE: We had some special "entertainment" on my BA flight in the form of a PAX who drank too much, took some pills and then went slightly off his rocker...ended up exposing himself 3 times. No, I'm not joking...cue the Chicago PD and EMTs when we landed. Edge to AA here...no crazy pax)
6. Overall, I'd give the edge to BA simply because I like flying the 747 so much more than I like the 777. I hear the NF on BA is great but can't comment on that sadly.