Presume or Assume?

[VladmirsPoutine]

Is there some mind-blowing nuance that I've missed with the use of these words? Surely they're interchangeable:

"I assumed you'd be there at 5."

"I presumed you'd be there at 5."

Both sentences infer that I thought you'd be there at 5.

What am I missing?

I Think they imply it and you infer it

en.oxforddictionaries.com/usage/assume-or-presume

When you presume, you do it based upon evidence.

When you assume, you do it without evidence.

Aka, if someone you worked with usually went to the gym at lunch time, you would presume that was where they were based upon their usual routine. You could assume that they were off work sick, however this would not be based on evidence.