Black-Box Zero Knowledge vs Non-Black-Box Zero Knowledge
The difference is to do with the quantifiers on the simulator. Non black-box zero-knowledge says that for all verifiers, there exists a simulator that can simulate a proof of all statements (i.e. there can be one simulator per verifier). Black-box zero knowledge says that there exists a simulator that can simulate a proof of all statements over all verifiers (i.e. there is one simulator total). Therefore, black-box zero-knowledge is the stronger definition of security.

Reference: Lysyanskaya, 2002 (page 31) https://dspace.mit.edu/handle/1721.1/29271