At the PhD level, International Relations is one of the main subfields of Political Science (the others being Theory, Comparative Politics, and American politics). IR focuses on the interaction between states (and non state actors, and more...). Political Science is a mainstream branch of academia filed under social sciences (along with Economics and Sociology etc). People get a PhD in PS typically to become professors.
Not to be confused with graduate schools of International Relations/Affairs/Diplomacy/whatever such as those offered by SAIS/Georgetown/Columbia SIPA/Tufts Fletcher, etc. Usually offering an MA or MS degree. Those are professional degrees for people who want to work in government, international organizations, NGOs etc. I would liken them to an MBA. To the best of my understanding, they don't have to write theses, often do internships or even work full time while taking coursework part time. Emphasis more on networking and gaining some practical skills.
Public affairs seems to be often used for MA degrees similar to the professional degrees, but I believe at some schools you can get a PhD in Political Science focusing on Public Administration. Is that what you're thinking of? Either way I'd say Public Affairs tends to focus more on the actual workings of government, like figuring out how to provide better public education or regulate businesses etc.