They're near identical in ranking for this course, right next to each other on
www.thecompleteuniversityguide.co.uk/league-tables/rankings/politics Birmingham is slightly better there, including for entry standards, general graduate prospects, and quality of research. Same story for the universities as a whole, for average across all subjects. However, Nottingham has a higher percentage of staff (68%) across all subjects on average conducting high quality research compared to Birmingham (63%).
For student satisfaction they're completely tied on 74% for this subject and 76% for the universities as a whole.
As a university in general, Birmingham has a slightly higher world rating (84th in QS University rankings. Nottingham is 100th) but they're less specific for Politics itself, stating just that both are in the top 101-150.
Birmingham is very much a city centre Victorian style/turn of the century redbrick university. Birmingham also has a canal area.
Nottingham appears to have more of a diversity of buildings of different ages. Its main building is a striking white stone building in parkland.
I'd visit both (the university in general, the department, and the city) and see where you can imagine yourself spending 3 years. It looks to me that Nottingham easily has the best halls of residences though so that would likely sway me.