Searchable encryption is a form of encryption where you can search for ciphertexts whose decryption satisfying some condition without decrypting those ciphertexts and looking at the decryption. The end result is typically a collection of ciphertexts.
Sometimes a secret key is needed to create a "search token" for a given query.
Searchable encryption for limited types of search is often quite efficient, but it tends to leak information.
Fully homomorphic encryption (FHE) is a form of encryption where you can do (a limited form of) computations on the decryptions of ciphertexts without decrypting those ciphertexts and looking at the decryption. The end result is an encryption of the result of those computations.
You can use FHE to do searchable encryption in various ways. This typically results in a very flexible form of searchable encryption that does not leak much information, but also quite inefficient (since FHE is currently rather slow).