TheFreeDictionary.com gives examples. www.thefreedictionary.com/counterfeit www.thefreedictionary.com/fake The register of each word is different as well. You can check the etymology to understand the difference.

counterfeit / fake are close synonyms , as in " This handbag is a fake/counterfeit" A counterfeit handbag is one that is made to imitate/copy the real one. A fake handbag > This can mean like above , meaning "a not genuine handbag ( of a certain brand) But "fake " literally means not real or genuine . His smile is so fake > You can't say " His smile is so counterfeit"