Another possibility is AA is quoting a code-share and BA is quoting their code. Before the joint business agreement I had numerous situations where AA code-shares were significantly more expensive than booking on the BA or CX or whatever flight number for the same booking fare. And AA couldn't sell me anything other than the AA code-share.
I'm not sure if this changed with the recent alignment. You may ask the agent the exact flight number and whether it is a code share or not.