West Wendover, Nevada, is adjacent to Wendover, Utah.
Little Rock and North Little Rock
Two cities separated by the Arkansas River. They both have Mayors and city governments.
And East Windsor. South Windsor is due east of Windsor and East Windsor is due north of South Windsor.
While East Vancouver is part of Vancouver, BC, North Vancouver City, North Vancouver District, and West Vancouver are separate municipalities.
Good one. I thought of Windsor Locks to the north (where Bradley International Airport is) but blanked on East Windsor. I never hear about that town for some reason.
Elsewhere in Connecticut is Haddam and East Haddam.
There’s also an East Hampton, but Hampton, Connecticut is halfway across the state and has no connection to the former. The settlers of East Hampton, Connecticut instead named their town after their previous settlement of Eastham, Massachusetts. And Eastham, Massachusetts has a village on the west side of the town called North Eastham, Massachusetts. 
Back in Connecticut, there is a Lyme and East Lyme (and Old Lyme). Nice area…too bad it’s probably most well known today as the namesake for Lyme disease.
Elsewhere in Connecticut there are the neighboring towns of Easton and Weston. Helen Keller apparently spent the last years of her life in Easton, Connecticut.
There’s no difference between cities and towns in Rhode Island. The only distinction is for census classification. Both have mayors and councils.
A little bit of research tells me: South Pittsburg, TN was named for Pittsburgh, PA (I can’t explain the “h”). The Tennessee folks were expecting their town to become the iron capital of the region — as big as the yinzer town — but, for reasons, it didn’t happen.
Don’t forget the three Bridgewaters - Bridgewater, East Bridgewater, and West Bridgewater.
We also have Attleboro and North Attleborough, but South Attleboro is a neighborhood in Attleboro (and yes, the pretentious citizens of the northern version insist on the “ough” ending).
Then there’s Easton, which is north of Norton; but both are well south of Weston. Stoughton is kind of in between them but more east, if “Stough” is meant to mean south.
It’s the same for Weymouth, South Weymouth and East Weymouth, MA. All separate cities. But North Weymouth is a neighborhood in Weymouth.
On Long Island, there’s Southampton, Westhampton, and East Hampton. They are near each other. However, Northampton is two states away in Massachusetts.
There’s also Easthampton, Massachusetts, which is southwest of Northampton.
The lack of an H is explained thusly: the US Board on Geographic Names, when it first began, decided to standardize all the -burgh’s and -berg’s to -burg. Most towns knuckled under, but Pittsburgh PA had a lng running fight to keep their H. Eventually they won, but all the other Pittsburgh’s got their name changed.
They are?
In addition to the Providences we have towns East Greenwich and West Greenwich. Originally it was the town of Greenwich, then divided back in colonial times into the pair of towns.
Weymouth has four areas of town, East, North, South, and The Landing. Four zip codes as well. But all one city.
It should also be noted that Vancouver and all of its municipal neighbors are entirely separate from Vancouver Island (which is not a city at all).
In the Interior, Kelowna and West Kelowna are separate municipalities.
Nelson and Fort Nelson are nowhere near each other.
And Hampton, NY is a few hundred miles away on the Vermont border.
Same story with Pittsburg, CA, steel and mining. The name of the original city wasn’t so much standardized until later, it’s the only one with an H.
See also Martha’s Vineyard, whose fighting spirit finally won out against the government’s ban on apostrophes in place names.
But there is also a South Boston, VA on the old Richmond & Danville Railroad. Virgil Caine almost certainly went through town at least once or twice, as he worked on the Danville train.
Maybe because people coming up the Tennessee River on steamboats ended up getting off at Pittsburg Landing, TN by mistake? Less iron but more history.