Though compliment and complement are related, that one-letter difference is enough to affect how you should be using each one.
As Merriam-Webster explains, they both have roots in the Latin verb complēre, “to complete.” But these days, complement embodies that meaning much more than compliment. To complement a given thing is to add something that completes or strengthens it. A choker necklace, for example, would complement a ’90s look nicely. You can express the same sentiment with the noun or adjective form of the word: A choker is a nice complement to any ’90s look; or a complementary choker takes your ’90s look to the next level.
Grammar also has all kinds of complements, which describe or rename another part of a sentence. The simplest ones are subject complements, linked to the subject via an aptly named linking verb. In the sentence The choker is black, the word black is a subject complement to choker.
Here’s an easy mnemonic device to help you remember when to deploy complement or its derivatives: Complement and complete both have two e’s.
If you tell someone you love their ’90s look, on the other hand, that’s a compliment. And if you’re handing out free chokers at a ’90s theme party, they’re complimentary (though, as established above, you could make the argument that they’re also complementary). Basically, any time you’re giving something to someone for free—praise, a hotel breakfast, etc.—you’d describe it as complimentary. Here’s a similar mnemonic device to keep that straight: Give and compliment both have an i.