Is there any difference between these two adjectives when they both refer to the USA? Can they be used interchangeably?
American currency/beauty/economy/flag/culture/way of life, etc.
US currency/beauty/economy/flag/culture/way of life, etc.
Ok, I will take a stab at this.
For currencies, we say: the euro, the dollar, the yen. Generally, when discussing currencies, we would specify the US dollar, the Canadian dollar or the Australian dollar.
The official name is: the US dollar, as in United States dollar. Where US acts as an adjective. The American dollar is used, also but not by traders.
Also, for the economy, both are used but, in formal texts, you'll usually see: The US economy grew [some percentage] year-to-date. The US economy is usually more formal and used in texts written by economists, in comparing it to other economies. However, more casually, you might see the American economy, too.
For flags, it can be either: the US flag or the American flag.
For culture, and other abstractions, generally, it would be American culture, Canadian culture, British culture, Chinese culture, etc. We generally do not use "US culture". Though we might find/say: Culture in the US etc.
Finally, for a word like beauty, it would be American beauty [like the movie], and not US. This is generally true of most nouns of this type. (abstract ones).
Finally, it is a happenstance of history that the word American is an adjective or a noun. This upsets some advocates of political correctness. All I can say about that is this: for every country in North, South, Central America and even the Caribbean, the country name can be made into the adjective and a noun from Argentina in the Southern Hemisphere to Canada in the Northern Hemisphere and all the wonderful islands of the Caribbean and small countries in Central America. This is not true of the country known as the United States of America. So, its citizens are left only with the possibility of calling themselves Americans.
Please note: in Spanish, they have a noun for a US citizen using the word United and States, and it's pretty cool: estadounidense. Unfortunately, in English, it doesn't work.
"The US" is a grammatically awkward country name. If it won't create confusion, it's usually preferable (speaking as a US citizen) to default to "American," which is just easier to construct sentences around. Depending on context, of course, this isn't always possible.
First of all, remember to include "the" with definite nouns: the US flag; the US economy.
It's not always that simple. Looking at your examples, "US" as a modifier sounds clumsy when referring to aspects of culture. The problem is, "the US" is a set of borders, not a cultural concept. While it's ok to refer to "culture in the US" in a somewhat generalized manner, it doesn't make sense to imply that US-ness is an inherent quality of anything, except the country itself. "______ in the US" is usually a safe way to go about phrasing this.
Does that mean it's okay to refer to these concepts as "American" instead? Moreso, but it will probably be perceived as old-fashioned.