In some cases, the two can be taken almost as synonyms. An assumption is essentially a guess taken to be true without further investigation or proof. For instance: I can make an assumption about the preparedness of my students.
Similarly, a presumption is an idea that is taken for granted (taken to be true) without certainty; however, the connotation for this word is that it becomes the basis for some added decision or conjecture. For instance: In our country, there is a legal presumption of innocence until one is proven guilty. This presumption must be the starting point of a jury's deliberations.
However, there are distinct secondary meanings to these words that are not so close.
Assumption can also mean the act of taking on a characteristic, an attitude, a power, or a responsibility, duty, or role.
Presumption can also mean a behavior that is (or is taken) as arrogant or disrespectful.