Is one more formal then the other?
Teach is including a more general concept.
According to Longman Dictionary of Contemporary English
someone whose job is to teach, especially in a school
American English: a teacher at a university or college
British English: a teacher of the highest rank in a university department
To see academic degrees visit here.
Also you may find it interesting to know:
teacher someone who teaches as their job, especially in a school : a high school teacher
principal ( also headteacher British English ) the teacher who is in charge of a school or college : The teacher sent him to the principal’s office.
tutor someone who gives private lessons to one student or a small group of students. In Britain, a tutor is also a teacher in a university : They hired a tutor to help him with his English. | Your tutor will help you find a subject for your essay. lecturer someone who teaches in a university or college : University lecturers aren’t very well paid.
professor a teacher in a college or university. In Britain, a professor is a high-ranking university teacher, especially one who is head of a department : She was professor of linguistics at Cambridge University.
instructor someone who teaches a sport or a practical skill such as swimming or driving : He works as a ski instructor in the winter. | a driving instructor
coach someone who helps a person or team improve in a sport : a professional tennis coach
educator especially American English formal someone whose job involves teaching people, or someone who is an expert on education : Most educators agree that class sizes are still too big.
trainer someone who teaches people particular skills, especially the skills they need to do a job : a teacher trainer | Many companies pay outside trainers to teach management skills to their staff.
governess a woman who lived with a family and taught their children in past times : As a governess, Charlotte Brontë received twenty pounds a year.
Professor is also a rank, and a title, in the American university system and as such, confers status on its recipients. There are lots of "instructors" and "lecturers" and "graduate assistants" who teach and publish original research who would love to be called "professor." So yes, "professor" is more formal than "teacher." In any sort of professional setting, for example in asking for the referral of an article or seeking university employment, I would be very careful to refer to the professor I'm mailing as "Professor" and not "teacher", so as not to seem overly familiar.