Both are pretty mediocre and not exactly something you should be thrilled about, but I will mention some unique differences.

BA PE 777
No dedicated PE lavatories
Cloth seats which are harder to clean and more prone to have stains from previous passengers
No individual air vents
Cheap blanket, pillow, and economy-style headset
Horrible food both in terms of quality and quantity
Often, very slow service since the majority of the cabin crew is assigned to Club World and there are relatively few crew to serve both Y and PE

AA PE 787
Dedicated PE lavatories
Leather seats and the bulkhead row has full leg rests
Individual air vents (available to all classes on AA; however I'm mentioning them given that BA lacks them in all classes)
Comfortable Casper blanket and pillow; better headset (not noise cancelling but better than the regular economy ones)
Slightly better food (particularly in terms of quantity) and presentation is also nicer since AA serves all items in proper china for PE
Whilst AA has the same cabin crew for both Y and PE, in general, I find them to be far more efficient than BA's and on many flights, I have gotten multiple offers for drink refills within a few minutes of receiving my meal