I’m a philosophy-polisci major at UChicago who’s planning on going to grad school, so I can talk about the strengths of the universities in your interests. UChicago undoubtedly has a stronger political science department across the board, including in political theory, which it’s top 3 in the world for. Brown and Chicago have fairly similar philosophy departments in general, though Brown is significantly stronger in political philosophy. I think you’re asking the right sorts of questions - if I was making the same choice as you, it would probably come down to the Core. I can say that the professors in both the philosophy and the political science departments, especially the younger ones, are fantastic teachers and care a lot about their students in my experience. The really famous ones (Mearsheimer, Nussbaum, etc) don’t interact as much with undergrads, but in general the profs are very accessible.
A big difference - Brown has notoriously bad grade inflation (possibly the worst in the country), while Chicago has notoriously bad grade deflation. Chicago’s rep is overblown and I think Brown’s likely is too, but it’s something to consider. Chicago students and faculty are also higher-quality than Brown’s in general due to its prestige and location.