I believe that on routes that AA shares with BA and IB, bottom-line revenue fares are the same on all three carriers. The only difference I'm aware of is that BA charges YQ on award tickets while AA does not.

And this is not the best time in history to spend FF miles (on OW TATL at least) - bankrupt and/or struggling airlines need revenue and cost savings more than they need loyal FF-club members and thus BA doesn't mind charging large sums for a 'free' ticket, and AA is happy when every seat is filled with a revenue butt so availability has dropped from 'generous' to 'grudging'.

For those without a loyalty account replete with miles (or 'avios') it's probably best to simply buy tickets on the cheapest reliable carrier. The few miles you pass up would be useful only if for a magazine subscription anyway.