Both "entender" and "comprender" are synonymous in most occasions, though "entender" has more meanings than "comprender.
However, there is a very slight difference between them. That difference is more psychological or philosophical than merely semantic. When you use "comprender" in some circumstances, you share a part from you with your speaker. For instance, I could say:
Te entiendo pero no te comprendo: I understand your words, but I don't share your opinion.
I hope you'll understand me, though maybe my thoughts
no son comprendidos 