Both words have numerous definitions in the OED, but in ordinary usage, both assume and presume mean “suppose.”
I suppose you are going to the beach this summer.
I assume you are going to the beach this summer
I presume you are going to the beach this summer.
H.W. Fowler’s opinion was that in using presume, the speaker believes the supposition is true and will believe it until he learns otherwise. In using assume, the speaker feels no certainty that his supposition is true or not.
In a legal context, presume means “to take as proved until contrary evidence is presented.” Ex. The defendant is presumed innocent.
Because of the association of the word presume with legal contexts, it carries a connotation of formality. For the fiction writer, presume would be the preferable choice in the speech of a remote or officious character.
Here are some quotations from newspapers:
… can’t even agree on why we disagree about President Trump, a USA TODAY/Suffolk Poll finds, but we assume the worst about the other side. Republicans described Trump’s opponents as lazy, narrow-minded and mired in … (www.usatoday.com)
… Even the most secure corporate networks tend not to take that sort of approach: once you’re in the secure zone, they assume you’re one of the good guys.” The future won’t be a hack-free heaven. Software is complex, mistakes … (www.theguardian.com)
As the heirs to those brave patriots who fought the American Revolution, we must not retreat from the ramparts they built. Due process was costly to achieve in battle and ought not to be conveniently ignored. When someone raises an allegation that the law has been violated, we must presume that the claim lacks merit until evidence is tested in a court of law and a judge or jury makes a ruling.