A marriage license is a piece of paper that authorizes you to get married and a marriage certificate is a document that proves you are married.
Typically, couples obtain a marriage license, hold the wedding ceremony, and then have the person who performed the ceremony file a marriage certificate in the appropriate county office within a few days. (This may be the office of the county clerk, recorder or registrar, depending on where you live.) The married couple will be sent a certified copy of the marriage certificate within a few weeks after the marriage ceremony.
Most states require both spouses, along with the person who officiated and one or two witnesses, to sign the marriage certificate; often this is done just after the ceremony.