As a current BA frequent flier (formerly very frequent on AA amongst others), I think it's a bit more complex of a question than it's being made out to be.

It's important to know what class of service you'll be flying.

For example, 5 sectors in business class from JFK-LHR will net you Silver status on BA (OW Sapphire, equivalent to AA PLT). The same flights credited to AA will make you an AA GLD (OW Ruby, equivalent to BA Bronze). Once you've hit BA Silver you earn 100% bonus redeemable miles.

As others have said, BA miles are now great for short-haul (domestic US or regional European) travel. For pure long-haul redemptions, AA is the clear winner. For upgrades, if you're booking premium economy or higher, it's a bit of a toss-up but BA works very well for me currently (long-haul upgrades cost 10k miles per segment for Y+ to J, 20k per segment from J to F, and BA's discount Y+ is usually similarly priced to upgradable AA Y fares).

As you mention, the AMEX transferability of points is nice. The AA and BA Visa cards are more of a toss-up, though if you plan on using your credit cards abroad the BA card has no foreign transaction fees.

AA's telephone customer service (especially for FFP-related issues) is miles (weak pun) ahead of BA's. My biggest complaint with BA is the lack of accessibility to quality customer service for anything mileage related. PM or ask on the board if you have questions about this point (or any other) since I'd rather not belabor unnecessarily.

Hope this helps a bit.

ETA: welcome to FT ^