The commonly-espoused difference between the MPA and the MPP, which I think holds true, is that the MPA is a degree for people who are primarily seeking to become managers. It is geared toward helping you become the administrator/director of a government organization or nonprofit. An MPP is designed to helped you become a policy analyst. There is some blurring between the two, and some institutions use slightly different terms for the degrees.
I am less familiar with MBAs with nonprofit concentrations, but as a "business degree for nonprofits," they are closer in spirit, at least, to the MPA.