The Merriam-Webster dictionary states for "assume":
to take as granted or true
And for "presume":
2> to expect or assume especially with confidence
3> to suppose to be true without proof
So although, the words are mostly used interchangeably, from these definitions, someone who is "presuming" something is more confident than someone who is "assuming" something.
My personal experience is otherwise, I have always thought of "presuming" something as assuming something prematurely, while "assuming" something would tend to have a stronger basis (though still a not very strong one).
For instance, I would say that if my pen was stolen, I would presume it was the first person that came into contact with me, if I thought he was averting his gaze. Whereas if someone had stolen something from me before, I would assume it was him.
Suppose is supposed to be a very near synonym to both words, but in my experience, it tends to mean that someone is a bit more hesitant at accepting something, e.g. "I suppose you are right". Or if my wife suggested I take out the garbage more often, I might say "I suppose that's what I should do."
I am very interested in
hearing reading other people's opinions on the matter.