One advantage not mentioned before that applies only if you travel within the US through cities where AA has flagship lounges, is the access to AA flagship lounges.
For me it's a great advantage as I fly through LAX 3 times a month. The flagship lounge is way better, having real food offerings, and being less crowded than the Admiral's club.
There are flagship lounges only at LAX, JFK and ORD, so this benefit is not significant for most.