BA and AA sell and rebook the same flights and can code over each other in such a way that in principle at least it makes no difference to cost or outcome. There is a difference on codeshare versus prime tier points due to coding issues but both airlines can handle the various workarounds. It's more likely to require a call in the case of BA. But in the round it's equal. Some FTers appear to like having the details visible in BA.com but that seems a marginal and perhaps wrongly comforting feature.

If you are booking a long way off, say more more than 3 months off, I see considerable advantage in booking AA. The reason being is that no airline shakes around schedules more than AA, it's quite amazing the retimings they do, I'd guess they do it 10 times more frequently than BA, to the extent that if booking 8 months out you can assume you will be retimed at some point. The upshot of this is that at least the AA telephone agents are totally aware of this, and when you call them with a sensible alternative (never call them without it!) then they'll have you rebooked in just a few minutes.

BA of course can rebook you, but they tend to need to contact AA for certain changes, BA agents are ultra cautious or unfamiliar about the whole process, many are even unaware that there is a liaison desk available for these changes.

One difference that may be a consideration if you are travelling at a bad weather time: if BA plated (125 ticket) and you get irrop'd into a lower class, anecdotally you are more likely to get the Original Routing Credit than on a 001 plated ticket.