The Local Police Station is just that -- it monitors the people in their homes and in their neighborhoods, is responsible for the overall neighborhood to which it is assigned.  The PSB is akin to God himself (or herself).  The PSB handles the various visas and resident permit issues, the more nefarious issues plaguing the country, etc., etc.

Basically, the Local Police Station is just there to see if your dog has is registered, process local robbery cases (nothing involving State Security).

The Gonganbu is responsible for the public security and anything by wide and large that involves public security : criminal investigation, terrorist activities, maintaining social order (in case of riot, demonstration), traffic regulation, household registry, identity card, passport, security on national borders, etc...