Any card associated with a specific airline will get you additional miles per rental. That's about it and that's about all your AmEx card gets you compared to any others out there.
Each card type (AmEx, MasterCard, or Visa) is pretty much the same in their basic offerings of rental car insurance. Some exclude specific countries and/or specific vehicle types. Most require you to decline all offered insurance from the rental company or they cover nothing. The local rental agency may require a letter from your credit card stating what they cover and how much before letting you drive off without their insurance. You will have to look at the coverage documentation provided with your card to know these details, or call them and ask to be sure.
AmEx offers a primary rental insurance as mentioned by others, but this is not included by just having the card. You must sign up for it separately and from the point you get it is it an approximately $20 charge for every car rental you put on that card. This does save money over the daily charge that many auto rental companies in the US have which can be that much or more per day.
I prefer not to rent outside the US and Canada simply because of the confusing traffic laws compared to home as well as the high price of gas, parking, and the ease at which many seem to get traffic tickets. I take trains or buses and have not yet found anywhere I wanted to go that I couldn't get to. It is also just more relaxing to me to not be burdened with keeping track of the car. But that's just me.