USA is an abbreviation for the United States of America. US tends to be the more popular way to abbreviate United States. Both of these abbreviations can serve as either adjectives or nouns. However, some people prefer United States as the noun form and US as the adjective form.
USA is a noun because United States of America is a noun which it is short for. Remember a noun is a person, place or thing. For example, "Don't forget that Hawaii is part of the USA."
US or U.S.
I was confused about this one for a long time but I read that either way is okay and it just depends on your style if you want to include the periods or not. I for one like to use U.S. I don't know, I guess I am just a true blooded American at heart and using the periods make it look more important. Ha Ha Ha...
US or U.S. can be used as an adjective to modify a noun. For example, "U.S. portions at restaurants are bigger than the portions in Japan."
Some people say that US shouldn't be used as a noun but instead United States should be used when a noun is needed. For example, "Yuka loved US culture so much that she decided to move to the United States". In this case US is an adjective that modifies "culture" and United States acts as a noun.
My conclusion is that whatever way you decide to use US and/or United States just make sure to be consistent throughout your writing. Don't skip around and use them in different ways.