Having flown BA economy on an AA ticket:

1. No difference in service, entitlement, or welcome;

2. Some categories of ticket will earn fewer Avios and/or Tier Points on AA but these are generally just the cheap, but not very cheapest economy fares - probably more expensive than the ones you're booking;

3. My inbound connecting flight was delayed and I missed the connection. I had plenty of time to get rebooked but BA didn't necessarily make the mental leap between AA xxxx and BA xx and the cabin crew, at least, assumed I'd be flying on AA metal for the long haul leg. If one delay meant rebooking was tight, that could be a problem for you.

4. As for the difference in price - either BA didn't load the sale fare, or it has sold out, wheres AA still has availability. It happens from time to time and isn't a big deal. I'd happily book the AA ticket.