dollar bill vs government note

dollar bill and government note both are nouns.

Word	Noun	Adjective	Verb	Adverb
dollar bill	Yes	No	No	No
government note	Yes	No	No	No

As nouns, government note is a hypernym of dollar bill; that is, government note is a word with a broader meaning than dollar bill:

dollar bill: a piece of paper money worth one dollar
government note: a piece of paper money (especially one issued by a central bank)

Other hypernyms of dollar bill include bank bill, bank note, banker's bill, banknote, bill, Federal Reserve note, greenback, note.

dollar bill (noun)	government note (noun)
a piece of paper money worth one dollar	a piece of paper money (especially one issued by a central bank)