I am AA Platinum and MPC Diamond.
Miles accrual: The AA elite bonuses are significant if flying on AA though AA doesn't give credit for mid-discount and all deep-discount CX fares. CX gives credit for almost anything on AA but with only minimal class of service bonuses. CX miles expire whereas AA doesn't if you keep flying
Using miles: CX miles are useful on CX and transpacific upgrades can be done with miles alone (no cash) from reasonably discounted fare classes. AA's MileSaaver fares are cheap and easy to book.
Free upgrades: I have had a lot of success with op-ups on CX, especially since the downturn shifted demand to the back of the plane leading to oversold economy. CX diamonds have first priority for opups and I have gotten several free transpacific upgrades from this. However, CX will only give free upgrades for operational reasons. AA will give lots of free upgrades with thier coupon system for domestic flights and eVIPs if you are Exec Plt.
Recognition: CX treats AAdvantge elites far better than AA treats MPC elites. CX will up up OW Emerald and Sapphires (behind MPC members, but it is still a free upgrade if you get it) and the staff thanks you for your business. You can also access the CX First lounge and even the CX arrival lounge with an AA ExecPlatinum card. When I fly AA with my MPC # I can't even select premium seat locations online and have to be firm with AA staff on the most basic OW privlages like lounge access and first/business class check in.
Getting and maintining elite:
CX has a complex system where the counter is reset to zero after reaching each level. This makes for a long trek to get to CX Diamond. MPC requires 20-50% more flying to achieve comparable one world status vs. AAdvantage once you factor in the higher thresholds and lower class of service bonuses.
Guaranteed availability (the real gem of MPC):
If you pay for your own tickets in transpacific economy the guaranteed availability in Y,B,H,K,L,M,V fares on CX/KA with as little as 72 hours notice (Gold) or 24 hours notice (Diamond) is very valuable. I save several thousand US$ a year buying long haul economy tickets on CX with this benefit vs. the quoted price on CX.com. AA has nothing like this.