The only difference would be in how you were using the two terms.
A member must show his or her membership card to enter the club. A member may bring one guest with him or her. If the guest misbehaves in the club, the member's card will be revoked.
I would not use member's card as a general noun in place of the more commonly used membership card, but that's just me and the dialect I speak. There is no rule about which is more correct.