I am a native speaker. I've looked at five different dictionaries and am now only a little less confused than you. Moreover, I think that the definitional entry from the Macquarie dictionary indicates how close the two meanings often are!
Presuppose. 1. to suppose or assume beforehand; to take for granted in advance. 2. (of a thing) to require or imply as an antecedent condition.
Nonetheless, despite the closeness of meaning, I think that presuppose implies that there was indeed some active thinking on the part of the presupposer. That isn't, so far as I can see, necessarily true of assume.
Consider, for example, a government financial forecast that is criticized on the basis of relying on "hidden assumptions". To me, there is no inherent implication in the term "hidden assumptions" that those assumptions (i.e., all of the necessary and sufficient conditions for the forecast to be accurate) were ever actively considered by the forecasters. In contrast, I don't think that presuppose could be used with the same meaning. "Hidden presuppositions" would suggest that there was something that was actively considered by the forecasters and then was deliberately hidden in the published analysis.