Wait, what ? Durarara!! is about as gritty as Kamisama no Memochou, if not more. It's an exploration of the seedy underbelly of Ikebukuro, with all that entails (gang wars, immigrant slums, human trafficking...). And we also get the same kind of dark introspective character explorations (I'm looking at you, episode #2 !).(And it reminds me one day I really ought watch Durarara!! so I can see the 'non-gritty' version.)
The main difference (besides Durarara!!'s writing being in a different league altogether) is that it's urban fantasy (with several supernatural characters wandering around), while Kamisama no Memochou seems more "realistic".