Roma are the city club, Lazio the surrounding countryside, ie urban working class for Roma, wealthy suburbans, land owners, farmers etc for Lazio.
Of course that's a generalisation, just like the fact that Roma are a left wing club and Lazio are a right wing club. Clubs like Roma have supporters who span the whole political spectrum and you don't have to go far to find extreme politics anywhere in Italy, Communism and Fascism still have a hold on Italian politics.