In common usage both assume and presume can mean ‘suppose’ and are often interchangeable in this meaning. However, technically there is a subtle difference between the two where presume is to ‘suppose to be the case on the basis of probability’:

Two of the journalists went missing and are presumed dead.

Assume on the other hand is to ‘suppose to be the case without proof’:

I just assumed it to be the case.

Both words also share other meanings that can be summarised as ‘to take on oneself’. In this sense assume is generally used to describe taking on a role:

I assumed the role of the aggressor in the conversation.

Whereas presume is generally used when taking on an attitude:

He had presumed too great a level of familiarity.