You will pass this course provided that you submit all your assignments on time and participate in our debates. However, I will not accept late work unless it is accompanied by a doctor's note or a very persuasive handwritten apology.

Provided that means "on the condition that" or "only if," which perfectly sets a strict requirement for passing the course.

Unless means "except if" or "if not." The professor will not accept late work except if there is a doctor's note. Using "if" here would logically mean the professor punishes students for bringing a doctor's note!