I did both back in April, and would do BA without a doubt (mine was LHR-JFK with AA and JFK-LHR with BA). Had Flagship Dining access at JFK for an onward connection and while it's nice, Its not the same atmosphere as the CCR (albeit it could be argued the food is slightly better). Same champers in both with the LPGS, but the service onboard is night and day. Ground service at JFK will be the same i'd expect, as I think BA are using T8 from December onwards, so AA/BA will be using the same facilities for F pax.

Don't get me wrong - the AA crew were absolutely fine; seat was comfortable (albeit a bit wobbly) and the food was ok, but there was no refinement you'd expect for the difference between First and Business class. Other than the seat, i'd say there was no difference with the AA experience vs business class, and the bubbles they were serving in first (yes, First World Problems) was a budget brand (£22 a bottle) vs LPGS in BA.

As with all things, it comes down to what your preference is. If the price was the same between BA and AA, i'd go BA 10 times out of 10. If you get a significant cheaper deal for AA, then it would make more sense if you want the TPs etc, but the crew on my last BA flight were honestly the best i've had in over 400 flights and the service was flawless. yeah, it's not on par with Asian carriers levels of F service, but it's the best of the two in my view.

Also *cough* if I'm not being inappropriate, I have a video review of my experience in AA First here - (BA one still to be finished) - if it helps at all.