Assume and presume both mean to take something for granted as true (among their many other definitions). The difference is in the degree of certainty. A presumption is usually more authoritative than an assumption. To presume is to make an informed guess based on reasonable evidence, while to assume is to make a guess based on little or no evidence.
In this example, presume is the better word choice because the speaker’s guess is based on irrefutable evidence:
I told the host, whom I presume to be the owner judging from the picture on the website, that I had reservations. [SF Gate]
And in this example, the speaker states that he doesn’t know, which makes his guess a clear-cut assumption:
I assume police have at least interviewed them. I don’t know if they have or not. [Express]