On ex-EUs, I had issues with checking in for AA-coded BA flights and had to go to the check-in desk, which is obviously a nuisance if you're doing a tight back-to-back turnaround. As someone else mentioned earlier, it may be easier to select seats if BA is booked as BA. On the other hand, some people mentioned they're less worried about dropping the last leg if they credit to, and have status with, BA, and the ticket is on AA stock (as it shouldn't be BA's problem).