While most resources and a quick Google search will tell you the words can be used interchangeably now, it wasn’t always that way. I believe as writers we shouldn’t use words incorrectly as a professional just because everyone else does. As for in your personal relationships, that’s another topic for another time.
Merriam-webster.com lists presume as “to suppose to be true without proof” and lists the example of “presumed innocent until proved guilty.” My read on this is that presume is without complete evidence. Therefore, I presume the earth is round even though I can’t see it.
Assume, also according to merriam-webster.com means “to take as granted or true” additionally it lists “suppose” as a synonym. The example on the website is, “I assume he’ll be there.” I have no proof he will be there. The website goes further to say “assume often implies a justifiable motive rather than an intent to deceive.” Example: “assumed an air of cheerfulness around the patients.”
I think it boils down to this:
Assume – no proof
Presume – some proof
Is this clear as mud?