In the US, each state is just that, a sovereign state, not a province or district. Both Texas and Vermont were actually independent nations at one point in their history. We tend to be rather proud of our independence of states. Prior to our Civil War in the mid 1800's, this nation was often referred to as "These United States", signifying the independence of each state. After the war, the people saw the need to unite, and the phrase became "The United States" signifying more unity as a body.
As it is in Europe currently with the EU, each nation is sovereign, but they have joined forces to make the economy more stable, transactions easier and travel easier. Really, its similar to what we did 200 years ago. Sadly, along the way, each of our sovereign states allowed the federal government have a whole lot of power to dictate how they work and to control a lot of the social aspects of the states.
Just an interesting note, in most nations, the Federal Government is the highest ruler, and the districts or provinces are subordinate. In the US, the Federal Government is supposed to be subordinate to the states, but over the years that balance has changed. In our early years, the Congress was considered the most important of our ruling bodies, since it represented the people and the states. Now, we tend to see the President as more of a leader, perhaps because of increased communication.
I live in Connecticut, a tiny state near New York, and am very proud of my state (though not always of Her politics). We were one of the original 13 to break from England, and we have some great history to go with that. My in-laws come from Minnesota, in the center of the country, and will talk about "this great restaurant that has been here forever....almost 10 years! Their viewpoint changed a bit when I took them to my favorite restaurant, a tavern that was occupied by the British during our war of independence, and is still standing in much the same fashion today.