Teacher is a bit more formal than a tutor.
In the context of high school or college, a teacher is the one who is actually teaching you the class. In high school, you call teachers "Mr. [Last Name]" or "Ms. [Last Name]" and in college, you call teachers "Professor [Last Name]."
A tutor can be someone informal who helps outside of normal class time. Maybe you take supplemental classes with a tutor at night, for example. Or a tutor can even be your classmate, taking the same class as you, who really understands the material well. In any case, usually, teachers have a higher educational level than tutors.