Some of the trouble comes from being a federation of states, not a single country. "The States" refers to the collection of states, "The US" refers to the states that are united, or, more loosely, to the union of States.
The problem is that there's very, very, very many nations made up of multiple states. Even Belgium is made up out of 3 parts that could very well be called states. "The USA" is therefore the only unambiguous way to refer to "The United States of America", in fact it might better be "The United States of North America".
Ultimately, the USA have no name for there union! Except maybe just "America", after the continent it is the biggest nation in. It would also retrofit the use of "Americans", which probably used to refer to people that lived on the continent, but now refers to citizens of "The United States of (North) America".
Ultimately it doesn't matter. The US, The States, the USA, it all refers to the same thing in people's heads. Just use the word your audience expects you to, that'll work best.