US Army vs United States Army

US Army and United States Army both are nouns.

Word	Noun	Adjective	Verb	Adverb
US Army	Yes	No	No	No
United States Army	Yes	No	No	No

As nouns, US Army and United States Army are synonyms defined as:

US Army and United States Army: the army of the United States of America; the agency that organizes and trains soldiers for land warfare

Other synonyms of US Army include Army, U. S. Army, USA.

US Army (noun)	United States Army (noun)
the army of the United States of America; the agency that organizes and trains soldiers for land warfare	the army of the United States of America; the agency that organizes and trains soldiers for land warfare