Yes, they are different. Just like in English, they have different meanings. "Teacher" is usually used for lower level students, e.g. elementary school teacher, while "professor" is usually used for upper levels, e.g. college professor. I've also read on other threads that calling someone "professor" is a way to show respect and honor to them.
In English, we can use the word "teacher" for anyone who who teaches us how to do anything. "Susie was my teacher. She taught me how to tie my shoes." I wonder if Arabic uses the word "mu3allim" the same way.