They are different and do not have much in common. Assume can mean to suppose or imagine something to be true without any proof. "He assumed that his friends had already gone." "Let's assume for the moment that everybody involved wants peace." Assume can also mean to take on responsibilities. "He assumed the office of president of the university." "He assumed the role of parent." = He took on the responsibilities. Pretend is usually used when we imagine or act as if something is true when it actually isn't. "Let's pretend we're rich." "The boys pretended they were cowboys."