Assume and presume both mean to take something for granted as true. 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; to assume is to make a guess based on little or no evidence.
Examples
In this example, presume is correct 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]
Here are a few more examples to help illustrate the difference between assume and presume:
They assume that all independents are bundled neatly together ideologically between Republicans and Democrats. [Boston Globe]
We can presume, fairly I think, that if de Tocqueville and Jefferson were around today to see what corporate interests have done to the American jury, they’d be furious. [Atlantic]
As a general rule of thumb, I always assume that anything dismissed by the EU as a ‘myth’ is almost certainly true. [Daily Mail]
Three friends, (we presume they’re friends since we see them meeting in bars and playing video games although they seem to share nothing in common), each have despicable, terrible, horrible bosses. [CBC.ca]