Why does Vegas have a police department?

In America, law enforcement is generally divided into three distinct types.
Cities have police departments that are run by a Police Chief who answers to a Mayor or a City Council. Patrol officers are Police officers.
Rural areas usually have a Sheriff's Department where the Sheriff is elected and doesn't answer to the local civilian government. Patrol officers are called Deputy Sheriffs.
States employ State Police, mainly for highway patrol and to guard State property. Patrol officers are usually referred to as Troopers.

Las Vegas has a police department, but it doesn't have a police chief. Instead, it has an elected Sheriff. The Mayor of Las Vegas has no official role with the PD.
The Metro Police cover the entire Clark County, all the way down to Laughlin, except for a couple of smaller departments such as Henderson and the City of North Las Vegas.

I'm not aware of another police force whose head is a Sheriff, nor can I think of a county that has a police department. Thats not to say there are none, but it is a very unusual situation.
Las Vegas actually has a separate Sheriff's Department that handles the city's jails and correctional facilities

If you lived in NJ or NY other than NYC, you most definitely lived in a county with a police department. Whether or not they call the head of it a sheriff is up to them. Being elected officials (usually) sheriffs have a lot of independence and it usually takes an act of the state legislature similar to impeaching an office holder to remove one. It used to be any citizen was eligible to serve as sheriff but many places now have laws where you must be a police academy graduate and sworn officer to serve as a sheriff or deputy, because they were electing people who didn't know what body part handcuffs go on or what end of a gun the bullet comes out of.

It mostly depends on whether or not a town is incorporated and if they're not it's next to impossible for them to hire police. Paradise NV despite having all of the Strip save the Strat, has as its government one unpaid, non-voting representative at the county commission. I believe all counties are incorporated. There are also other large institutions that hire their own police forces; universities, airports, and at least one instance where a church had to hire their own police because they had a huge congregation and happened to be located within a little town that did not have the police resources to deal with their traffic. Amtrak at least used to have their own police force with an interstate commission.

Many deputy sheriffs get assigned to work as court bailiffs.

Don't forget game wardens and postal inspectors!

Nassau County has a police department run by a Police Commissioner appointed by the County Executive. Nassau was once part of Queens and was policed by what is now the NYPD. When it split from New York, it kept the police department. It is the newest county in New York State. Suffolk County, next door, is one of the oldest counties in NY State and only recently established a county-wide police force. Before that, the various towns had police forces, and the State Police handled the Eastern Forks. Suffolk has a Sheriff who runs the Sheriff's Department and a Police Commissioner who runs the police. Suffolk's entire government still uses procedures established in Colonial days.
Westchester County doesn't have a county police department; it has a Department of Public Safety. There is a difference, although I don't know what it is.

Having an elected sheriff as the head of a police department is very unusual, as far as I can tell. Police Departments are run by appointed Commisioneers, answerable to a mayor or council.