Police v Sheriff

Both are law enforcement types. Police are hired for a specific city or town. Sheriffs get county wide enforcement. I don't know if there are jurisdictional issues or not. So far as I know, all sheriffs are elected, and there is only one per county. Sheriffs have numerous deputies.

Were you looking for something more specific?

Question arose rather stupidly. Was watching Sons of Anarchy. In one Ep, the city council decides to get rid of the local PD, and put policing under xxxx county Sheriffs dept.
Town v County?

That would explain it. Matter of fact, I heard of such a situation in the past half year in which the town police were running a blatant speed trap, but only stopping drivers from out of state. I don't think the town council was given a choice, in that particular.

To go a bit farther, I expect state police to be the most professional of all. As I said, sheriffs are elected and require financial support for elections. City police can be very good, or not good at all. All that being beyond the scope of your question.

How do the various members of each group get trained?

Training for police officer varies from one state to another. So, for example, in Illinois, until the late 1970s, there were no standards for the training of police officers. In about 1978 or 1979, legislation was passed which required training minimums for any officers who had not gone through the State Police training academy, or the Chicago Police training academy. The PTI--Police Training Institute--was set up at the University of Illinois. (I happened to be employed at the University of Illinois at the time, and many of these officers frequented a bar where my brother worked. Initially, they were veteran policemen who already had time on the street. Later, they were new officers with no experience. The difference was enormous--the latter group were a bunch of loud-mouthed cowboys, and caused a lot of trouble.)

The spread of SWAT teams (Special Weapons and Tactics) lead to calls for more training of police officers. This is even more crucial as more and more surplus military equipment is just handed out to police departments. I long lived in Columbus, Ohio, at a time when Ohio did not have police training standards. There was the Columbus Police Department, the Franklin County Sherrif's Police and more than a dozen small municipal police forces. As Columbus grew and swallowed up small enclaves within Franklin Country, the police forces and fire departments were left in place, so that they would be paid for by local taxes. It only got better when more and more complaints came in about speed traps and police harassment by these cop shops scattered around Columbus.