Ventajas
The work environment in my last 2 groups has been great (Risk and Service Engineering). Overall, a great workplace (the one exception that I know of being the call center, and even that isn't too bad compared to other call centers). The "work-life balance" is good (a la, they typically don't expect you to work much more than 40 - 45 hours a week. Vacation time is reasonably plentiful, etc. Also, it is a very good place to work, and while it is difficult to get the large promotions from within, it is very possible (as compared to other companies, where I would be surprised if it were less possible).
Desventajas
I'm not sure, but I suspect that compensation is only "average". Also, like other large companies, there is a fair amount of bureaucracy, red tape, time wasted in meetings, etc. However, I don't think this is specific to only American Express, but rather, all large companies.