I looked up the words in my Arabic/English dictionary. "Professor" is "ustaadh" but "teacher" is listed as "mudarris". I find having a dictionary helps a lot. I use the "Arabic Practical Dictionary" by N. Awde and K. Smith. It has an easy to use layout.
All professors are teachers, but not all teachers are professors. A teacher is anyone who teaches us anything, our parents are our first teachers. A professor is someone of the highest academic rank who teaches at a college.