Usually, the right word is "American." "America's" is rare. The meanings overlap, they are neither exactly the same nor completely different. "American" is an adjective, and it means we're thinking of "American" as characteristic of something. Parallel examples: "a charming accent," "a slight accent," "an American accent." Similarly, "the large flag," "the beautiful flag," "the American flag." Other examples of usage: "the American Dream," "the National Museum of American History," "American popular music." "America's" is used rarely. It's the possessive form of the noun America. We use it when we are personifying America. The speaker is thinking about America's national identity, as if America were like a person (and Canada, China etc. were other people). It emphasizes something belonging to America, rather than the American nature of something. Just as we can say "my funniest home videos," "Susan's funniest home videos," a popular TV program calls itself "America's Funniest Home Videos." Similarly, "England's symbol is John Bull, France's is Marianne, and America's is Uncle Sam." "American" is a plain statement of fact. "America's" conveys a sense of the United States as a person. It is often used in statements involving patriotism or national pride. (Be aware that some Latin American dislike the U.S. use of "American" to refer to the United States of America).

I would say the difference is when you are talking about America, and what America has, it is "America's". America's mountains - we are talking about America, and the mountains it has. When you are talking about things that either are in, or from America, or have the nature of "America", if you like, it is "American". American mountains - we are talking about mountains. Which mountains? The ones in America. There are a lot of times when either could be used, but I would say that is the main difference between the two.

Well,I guess that will give rise to another question as to the difference between "of" and " 's".Haha,I know they are used in almost the same circumstances.

Iam waiting for Native speakers answers because I searched it and found different explanations. My humble point of view, I see the topic depends to what you want to express or say whether you want to say these things or people are ogionally American and belong to America as an adjective as in American cars,citizens.....etc , or you want to express ownership as in: America's Bank Put in your mind you can also use 'of' as in United states of America