They're usually synonymous and interchangeable. However, mimic can be used as a noun to describe a person who is good at imitating things. When I hear mimic in speech I consider it to have a more negative connotation than imitate, but that might just be me.

Imitate is used more often than mimic and has a more general and neutral meaning. Imitating someone could be a good thing or a bad thing. Mimic almost always means you are imitating someone in order to make people laugh and/or to ridicule or humiliate the person.

Imitate implies following something as an example or model: "He imitated the playing style of his music teacher." Mimic suggests imitating someone's mannerisms/voice/etc for fun or ridicule: "They liked to mimic the teacher's funny accent."

Imitate is used more often than mimic and has a more general and neutral meaning. Imitating someone could be a good thing or a bad thing. Mimic almost always means you are imitating someone in order to make people laugh and/or to ridicule or humiliate the person.