A teacher assigns homework to students to do at home. An assignment is any task or duty that is assigned by anyone to anyone else. By the definition of "assignment", homework can also be called a "homework assignment" However, my boss at work gives me assignments because only teachers in school give homework.

Homework's objective is generally to improve student's knowledge on a certain topic to be completed at home, while assignment is to help students to tackle and understand the problems and its solutions.

homework > work given to a student to be done at home. assignments > work that is assign ( is given)to you ( student or anyone) to complete by someone ( most likely a person of higher authority ) His boss gives him the assignment to take care of the visiting customers . *homework assignment > assignments that you take home to do (for students)