Well, there are a few reasons as to why I'd take UA over AA.

Being Premier Executive, I get to enjoy RCC lounges on international tickets. AA frequent flyers at any level do not.

UA's mileage runs are much better than AA's - hands down - due to plentiful Asian and European routes. Not to mention they're consistently cheaper on UA than AA.

Award trips require lower mileage on UA than AA.

Harder to get to Premier Executive on UA than AA (Reason: AA has way too many million milers because they factor in bonus miles as well - effectively watering it down).

Only nice thing I can think about AA is that I don't have to scroll all the way down on FlyerTalk just to get to AA's message board. UA's all the way down.