These two words are pronounced the same, but they are spelled differently and have different meanings.
Compliment can be both a noun and a verb. A compliment is a positive comment about someone or something, for example: “You have beautiful eyes!”
And the verb compliment (or the expression “pay someone a compliment”) means to make a positive comment:
The adjective complimentary means “free”:
Complement can also be a noun or a verb. If two things complement each other, it means they go well together; they make each other appear better.
The adjective complementary means that two things are different, but go together well – you could say, “My business partner and I have complementary skills.”