In many situations - where the meaning is just 'to suppose' - the two words are effectively synonymous..ie mean pretty much the same thing. Otherwise, I wouldn't quite agree with what North Star has already suggested above.
It does depend on just how doubtful about the truth of the matter the speaker/writer is, but to assume means to take for granted as a basis for discussion (some doubt) and presume means just to take for granted or count on (no doubt).