The words can often be used interchangeably. They have subtly different meanings that you should keep in mind when using them -- but don't assume a speaker or writer who is using them understands the difference!
"Assume" implies a bit more finality than "presume." It is more appropriate when a decision has been made, or is about to be made, with incomplete information.
"We didn't know how many people would be at the event, so we assumed that 100 chairs would be enough. (the decision was made)
"Since you didn't reply to my earlier email, I assume that you will not be joining us at the meeting." (implies that the speaker is ready to act without additional feedback from the recipient)
"Presume" gives a bit more room for the listener to correct the speaker. Using "presume" instead of "assume" adds an implied "but correct me if I am wrong" to the end of a statement.
"I presume you want chicken for dinner[, but correct me if I am wrong]."
In the past tense, "presumed" carries a bit of humility (if referring to oneself) or judgment (if referring to someone else), and it may imply that the presumption was incorrect in hindsight.
"I presumed that she would not object[, but she did object]."
"They presumed to know my intentions[, but they did not]."
"Assumed" is more neutral, and does not imply anything about the outcome of the assumption. If you want to communicate that the assumption was correct or incorrect, that fact will need to be made explicit.
"I assumed that the dog would not bite. I was wrong."
"She assumed that there would be enough cookies, and she was right."
The following examples contrast the two words and explain their implications:
"You have not told me what kind of ice cream you want. I assume vanilla will be fine."
(Even if the listener says nothing, the speaker will give them vanilla ice cream)
"You have not told me what kind of ice cream you want. I presume vanilla will be fine."
(The speaker would like the listener to confirm or reject the presumption. They will probably wait for an answer before giving the listener any ice cream)
"Alice assumed I would like chocolate cake for my birthday."
(The speaker may or may not have liked chocolate cake. It would be risky to guess without more information)
"Alice presumed I would like chocolate cake for my birthday."
(This is a subtly judgmental statement. Alice probably got it wrong.)
All of this is based upon present day, American usage. The words could carry different connotations in other countries or in older usage.