"Paper money" suggests the physical currency, instead of using a bank card or credit card (plastic money). I think this phrase is going out of date quickly, seeing that (for example) my country uses polymer banknotes, and US dollars aren't technically paper, but cloth. "Banknote" means a larger currency than coins.

In the US, we don't use "bank note". It would be paper currency, or more commonly, paper money, or more commonly still, the amount combined with the word "bill": dollar bill, 20 dollar bill, etc,

From the Oxford dictionary:- paper money: Money in the form of banknotes. banknote: A piece of paper money, constituting a central bank’s promissory note to pay a stated sum to the bearer on demand. A circular explanation, more or less. The only difference that I can think of is that there are some kinds of paper money that are not legal tender, e.g., Monopoly money. Proper banknotes are issued only by banks (usually central banks) and are generally considered legal tender.

Depends a lot on the context.