Assume has a variety of meanings. It basically means "to suppose or take for granted," "to pretend," or "to be taken up." Presume is related to and similar to assume, but it has the sense of doing it beforehand. It means "to dare or venture without prior knowledge," "to assume as believable without direct proof," "to take as a premise, subject to further proof," or "to behave arrogantly or over confidently."

These words can be synonymous. You can interchange them if you mean to say "to believe something to be true." Ex. Because you were not here at 3pm, I presumed you will not arrive. Using "assumed" would also make the sentence correct. However, these words have other meanings. Assume can mean "to adopt a role" or "to take responsibility." Presume can mean "to take advantage".