They mean approximately the same thing, that is, to believe something to be true without knowing for sure. The primary difference is the degree of certainty of the speaker. When you presume something, you are more confident than when you assume something. For example, say you work for a restaurant, and a loyal customer comes in for dinner. You might ask, "the usual, I presume?" This is because you are quite confident that the customer usually orders the same thing. Say, on the other hand, you know nothing about a new customer who walked in. Based on her expensive attire, you thought she would order the most expensive dish on the menu, but instead she ordered the cheapest. You turn to your colleague and say, "I assumed she was loaded and was going to order the filet mignon. Instead she ordered the pasta dish!"

There is a different, but it is very minor. To presume is to make an informed guess based on reasonable evidence. To assume is to make a guess based on little or no evidence.