If you don't have any prior knowledge about the actual meaning of the two words, you might assume both are synonymous just because of both start with the same word, i.e., "Hold." However, there are little differences between the two. One of the several meanings of the hold up is to lift or support something probably with one's hands or anything.
For example: hold up the instrument so that everybody can see it. Also, it means to fulfill one's part of an agreement.
For example I believe John is holding up his side of the bargain. It also means to survive or to withstand. On the other hand, hold on means to grasp something in a firmly manner.
For example: Hold on to the chair so you won't fall. It also means to wait for a short while. For example: hold on for some minutes, I have got something important to do here.