This boils down to the chief difference in that the United Kingdom is a Monarchy and the United States is a Republic (In the sense that the Head of State is not a hereditary office. Americans also use the word "Republic" to mean "Representative Democracy" because they became a "Representative Democracy" before the term existed and the thing they wanted to model was the Roman Republic).
This meant they were also working with a system where the Parliamentary Equivalent was not running the Government of His/Her Majesty, but rather for the general population (Of the People, For the People, By the People). General Populations do not govern and since the duties of the Cabinet were removed from our Parliamentary Body (Congress) and given to the Head of State (Making the office of President Equivalent to both the full duties of the Monarch and the Full Duties of the PM). Thus, Congress is not governing, as they don't run stuff... they just make the rules, and the nation doesn't belong to a King, but an elected President (title derived because he Presides over the Cabinet in the stead of the Prime Minister/Speaker of the House), so he's got equal stake in the nation as any other citizen, the President is not governing but Administering the will of others (Congress through laws, the general population through elected mandate).
So where it might be the Churchill Government in the U.K. but in the U.S. it would be the Roosevelt Administration which are effectively the same thing... Either the current Head of Government or his Cabinet or Both.
When Americans refer to the Government, they generally are referring to the institutions of governance, not a particular elected officers' running of it. In it's most broadest sense, it encompasses the three branches as outlined in the Constitution (Legislative, Executive, Judicial) and their various components. Americans also have a general distrust of their own government and it's generally used more negatively than when referring to an Administration, which is mostly neutral though can be positive or negative in usage, depending on who's administration it is).
This is probably the longest bi-product from the Revolutionary War, in that, the catalysts were the unpopular restrictions put upon the Colonists in the days prior, initially to pay for a war that was partially fought defending them, but then as punishment for protesting those taxes in the form of the Tea Party. The amount of anger that would drive a British Citizen to destroy perfectly good Tea does not dissipate after just 200+ years of separation. Since they were at the time British and proud of it, they weren't by the British People, but the Government. Americans still see the Government as more of an all consuming three-headed monster that would gobble them up in an instant if not for the facts that each head will fight each other over who gets to eat the people.
This also could explain why terms developed away from the U.K. The Founding Fathers wanted to get rid of everything British from their government. Voting day in the U.S. is always Tuesday because when you eliminate all the days that were terrible days of the week for an election, you're left with Tuesdays and Thursdays... and it couldn't be Thursdays because that's when the British vote and we aren't like those tea sucking tyrants at all (There was also a serious movenment to write the Constitution in German and declare it the language of the land for no other reason than the Tyrant British speak English, and we aren't English. Thankfully, this was concluded to be too stupid, so we all agreed to call it Soccer and mispronounce Aluminum and called it a day.). Suffice to say the "Special Friendship" wasn't always a Friendship.
Edit to address comments made by @RinkStingpiece:
The word "Gubenatorial" is an adjective use to describe things related to the office of the Governor of a State similar to how "Presidential" when discussing things related to the office of the President (you typically hear it in election years as the Gubernatorial Race is used to distinguish from the Presidential Race). The use of Governor descends from the colonial states each having a Governor or Governor-General as described, though these days a Governor is basically the "President of a State". An individual governor's period in office is still called an "administration". The term does have roots in the British office of "Governor" or "Governor-General" in that they were stand-ins for the Crown who could provide royal ascent to the colonial legislature on more day to day matters of the colonial legal powers. Following the Revolution, each Colonly was effectively a separate nation-state that had a co-operation of a Confederate treaty government and later a Federal Government. So in this case, U.S. Governors are effectively "Presidents in miniature" and has all the powers of the President, sans those all states delegate to the Feds. It's better to think of the U.S. as like the E.U. and in fact, if you allow for the briefest moments of time, the U.S. includes 17 separate nations (the original states, as well as Florida, Texas, California, and Hawaii were all briefly independent before signing up with the U.S. States Hawaii is the only one that entered through annexation, and the rest voted (though Texas and Florida briefly wanted out)). And yes, each governor is the Commander and Chief of the state's national guard, though this may change in war time. Compare with U.K. governors, which are appointed by the Crown to act in his or her stead when communications with England aren't timely (the sun never set on the British Empire, but the queen's gotta sleep), or Governor Generals, which are used by former Colonial nations of the British Empire that still wanted the Queen of England as a head of state. The difference is that Governor-General are appointed by the Parliament (or whatever the hell Australia calls their Frankenstein of a Legislature). For Federal Nations like this (Canada and Australia) the provincial/state level equivalent of a Prime Minister is usually called a Premier.
The United State government specifically can not legislate on just anything and there is language in the Constitution that basically says that any power of Government that the Constitution does not say the Federal Government has, then that power devolves to the States and the individuals to determine... it also bars states from conducting any power of Government that the Constitution explicitly gives the Feds and later after a loophole lead to the Civil War, it was patched to also block states from any powers that were explicitly listed as things the Feds cannot due (Most of the Bill of Rights is written as "The Government shall not do this thing) and it's generally understood that only governments can break constitutional law.
What this means is that the the Governor of any U.S. State is a sovereign head of government/state with respect to the list of things they can do. Thus the U.S. Governor is not acting for the President within one state, but is a seperate power to the President. Whereas, a U.K. governor, or a governor-general's job is effectively "keeping the throne warm for the queen while she's in England and not Canada."
A lieutenant-governors is the Governor's equivalent to to a Vice President or whatever they call the Prime Minister's immediate successor (Secondary Minister? Nah, I kid, I think it's a Deputy Prime Minister). They basically need a pulse in case the Governor suddenly doesn't (This is a U.S. sentiment as the U.S. Vice Presidency has largely been seen as entirely useless unless the President suddenly becomes, as Monty Python would call it, an ex-President.
Finally, because a U.S. Citizen will typically be under the jurisdiction of no less than three different governments (Federal, State, and Local, though if in D.C. you can get down to two) and technically, could be under as many as Seven (though you have to be standing in one very specific spot in the country and commit a crime in a specific way), the "Government" typically refers to the Feds, while "the State" or "[State Name]" will refer to the State Government (i.e. If I am made about a specific state, I would say "Can you believe what California is making me do"), and the "[Local general term]" (i.e. "Can you believe what the County is making me do?" Substitutions of City, Town, or other approriate legal entity will do. Occasionally, if it's a major city, referring to the proper name is helpful.). Typically, using "The Feds" will only refer to Federal Law Enforcement, regardless of agency, with exception to the Military, which will be refereed to by the Branch. Typically, the FBI, DEA, and occasionally the TSA will get this, though many Americans will generally not give them the dignity of being the hated "Feds"... and compared to most interactions with the TSA, this is still a very dignified form of disrespect to them. The IRS are typically called "Revenuers" though the term is usually commonly shouted by any poor southerner when being swarmed by any number of men with guns, regardless of reason (and cause they commonly make moonshine for sale, but don't pay liquor taxes, which gives the impression that the government is coming cause it's not getting a cut of your illegal profits... not because your money was illegally made in the first place. The IRS does have one of the highest successful prosecution rates in the nation). By and large, most crime in the U.S. is handled at the state level, and the average person with a history with the law still may still never be prosecuted for Federal Crimes (though they almost always broke one if convicted by the State... the Federal Government rarely prosecutes a federal crime when the state has a similar state or local crime on the books). As I said earlier, U.S. Citizens tend to see the government as a necessary evil rather than on their side, and there is an amendment that specifically protects the right to call them all kinds of divisive names.
