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:
▶️ He complimented me on my web design skills.
▶️ He paid me a compliment on my web design skills.
The adjective complimentary means “free”:
▶️ At this hotel, breakfast is complimentary.
▶️ You get a complimentary cookie when you order coffee at that café.
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.
▶️ That necklace is the perfect complement to your outfit.
▶️ My business partner and I really complement each other. We make a good
team.
The adjective complementary means that two things are different, but go together
well – you could say, “My business partner and I have complementary skills.”