If it has an AA flight number it wouldn't be covered by price promise, and the t&c may well be different, but that doesn't mean you won't get a price promise if you run it through Momondo or similar and see if there is a better price available from an OTA.

I'd tend to book direct with BA, though I'm not sure why: AA online and phone support is superior generally, and it's a better price. I think I'm influenced by habit and status.