What is the difference between a forum and an assignment?

A forum is visible to the entire class or group. This is where you and other learners respond to questions or prompts provided by the instructor, and reply to posts made created by other learners. 

An assignment, sometimes referred to as a "dropbox", is private between you and your instructor. You submit your work by uploading or typing it into the assignment activity, just like they would hand in a paper during a class session. Your instructor will then be able to privately provide you with a grade and feedback.