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Prophets are clearly distinct from teachers. In basic terms, a prophet is one through whom revelation is given, while a teacher is one who explains revelation.
In Acts 13:1 we find that both prophets and teachers were ministering at Antioch. Note what Paul writes in 1 Corinthians 12:28-29:
28 And God has placed in the church first apostles, second prophets, third teachers, then miracles, gifts of healing, helps, gifts of leadership, different kinds of tongues. 29 Not all are apostles, are they? Not all are prophets, are they? Not all are teachers, are they? Not all perform miracles, do they? (1 Corinthians 12:28-29, NET Bible)
Prophets are clearly distinguished from teachers here.
Prophets are those through whom God reveals truth, while teachers are those who explain the truths of God's Word.
I think it is important to keep in mind what Deuteronomy 13:1-11 and 18:14-22 teach us about prophets. They are not to make mistakes. And they are not to contradict Scripture, either.
Could a prophet teach? I suppose so. I would suspect that Paul had both the gift of prophecy (it was, after all, through Paul that much revelation was given -- see, for example, Ephesians 3:1-12) and the gift of teaching (see Acts 20:7, 17-20, 27). Nevertheless, the two gifts are not the same (see also 1 Corinthians 14:23-26).