Basically, the function of a police officer is the same no matter where you live in the United states.
A county Sheriff in most northern states is relegated to running the county jail, court house and serving warrants and civil process. In the southern states, the county Sheriff is way more important and powerful. I’ve worked for the local Sheriff for over eight years, and I can assure you, he has way more powerful both in responsability and and politically. As deputy Sheriffs, we tend to leave law enforcement in the city limits to the police department. However, our authority extends to the city as well as the county. The authority of city police officers is only within city limits.
Our Sheriff and the local police chief have a very good relationship with each other and our two departments work very well together. Unfortunately, this isn’t always the case in all areas. Also, the police chief is appointed by and works for the city commissioners. They can hire and fire the chief at will. The Sheriff cannot be fired because the sheriff is a constitutional officer. The Sheriff must be impeached just like any elected official. That’s why it is very important to choose the Sheriff carefully, they’re not easy to get rid of.