They're 2 different security codes for cards. Amex has 2 levels of security: they have the normal CVV (Card Verification Value) and the 3 digits are a CID . CVV is a calculated highly secure 4 digit code based on your card number that is not contained in the card magnetic strip, which is the reason why it's printed on the card itself.
If someone were to take your magnetic strip or print your card. It would not contain your CVV or CID, so it could not be used for card not present transactions or most high-end merchants, which require CVV, CVV2, or CID as a form of further verification that the card is indeed there.
CID (Card Identification Data) on the back of the card is similar to Visa/MC/Discover. It's a 3 digit number that again is not stored on the card magnetic strip and is not visible on the front of the card, but again ensures that the card is in your possession at that the numbers have been not tampered with.
CVV and CID verification process of card numbers is not published or available for many, many reasons, so I have no idea how they verify your card number, but they do somehow.
To get to your question why does Amex ask for those 3 digits?
Because they can.
They require most merchants these days to verify the CVV before processing payment, but for their measures of security they verify the CVV and CID. I think most of us who have had the card for many years memorize the CVV, but I couldn't for the life of me memorize the CID, so that could be the reason to.