City manager would be in charge of city business (building and occupancy permits, inspections, billing for services, etc), whereas the Mayor (and/or city council) makes the political decisions (setting the policies the manager executes, etc)
It seems a bit silly to be incorporated with less than a thousand people, but I don't know the local history. If there's not that much to do, the city manager is likely part time, and the mayor probably does mayoral stuff a couple times a month. (Quick monthly meeting + maybe show up to a community event or three)
A 6-person department is just about the minimum for 24/7 coverage. IDK why they have a department at all though. Just let the state police or sheriff deal with it IMO.
One guess: response times that measure in minutes not hours.