What is the difference between to presume and to assume, as in "I presume you'll be at the party" vs. "I assume you'll be at the party"?

In that particular context, nothing I can see. But for me, as a mathematician, 'presume' has some factual meat behind the prediction, whereas we can assume absolutely anything at all. Assume I am the Pope; assume also I have four heads, and every head needs to be covered by a mitre, and that no blue mitre can cover more than one head at once, whereas a purple mitre can cover two, and a crimson mitre can cover three. Given these assumptions, we can work out what logically follows as the possible combinations of mitres that would suit my needs.

Shu, your question interests me. I did some research and found a pretty good (or so it seems to me) explanation of the difference between "presume" and "assume" at Grammatically Correct.
This part of the text made it clearest to me: "Hank presumes he’s the smartest one at the conference...Hank assumes he's the smartest one at the conference. Either answer will work, but each word creates a different attitude for Hank. If he presumes, we think he's overconfident or even arrogant. If he assumes, we ascribe a more humble attitude to him."
I was getting pretty comfortable with that until I began to wonder What if a presumption is made about a positive trait belonging to someone else? In other words, if I say that I presume Hank to be the smartest person at the conference, surely that would not make me arrogant???
Okay, folks, have at it!

I happened to come across a realistic example of this just this morning. It confirmed the idea that presumption is a kind of belief and is backed by evidence, whereas assumption is more a kind of action (if you distinguish them).

I had changed phone providers to a cheaper one so I presumed I didn't need to keep dialling the special access code to get the other one... it was a reasonable presumption, but I wasn't prepared to risk it and pay old-provider prices, so I wasn't going to assume it (that is, act on the assumption).