A “compliment” [per Webster’s New World Dictionary] is an expression of admiration, or a gift given for services (tip); a complimentary object is given free as a courtesy.
“Complement” is that which completes or brings to perfection. Complementary objects operate together, each providing what the other does not.
Examples of proper usage:
“My compliments to the chef for providing complementary proteins in our vegetarian lunch this afternoon.”
“The owner of the restaurant is a family friend: She sent over a complimentary bottle of wine which complemented our meal quite well.”