For the United States things are pretty limited, but generally the rights and privileges are going to be some sort of guarantee that the degree granting institution will verify that you were in fact awarded the degree in question. Since a lot of jobs now require degrees this is a non-trivial right. Additionally, alumni usually get some sort of perks such as library borrowing privileges, access to athletic facilities, etc. but that is very dependent upon the school.
Of varying use depending upon where you live you can use post-nominal initials to indicate the degree awarded and being addressed as "Doctor" if you were awarded a suitable degree. As noted in the answer by jackbael, you also have the academic right to wear the regalia of the school you gradated from as well.