“L’argent” can be related to everything about money like banknotes or coins.
“monnaie” is usually use to speak about coins only.

Argent (which literally means “silver”) is used to mean “money” in all cases.

Monnaie has several different usages:

• “coins”, especially petite monnaie

• “change”, as in “the money given back to you after you pay with more than the exact amount”, which can perfectly well include bills

• “any small amount of cash”, as in T'as de la monnaie ? (an English-speaker could well say “Do you have some change?”). I often understand this as stemming from “do you have some small coins so that I can make up the exact amount on what I have to pay so that I don’t have to break out a large bill or cheque or card?”, which is more polite than “I don’t have any money, can you help me out?”. I have absolutely no research for this interpretation!

• “currency”, as in La monnaie américaine est le dollar, or La monnaie britannique connaît des problèmes à cause du Brexit.

You also have the word pièce / pièces (short for pièces de monnaie), which (in context) always means “coin / coins” (cf. “thirty pieces of silver”, “pieces of eight”), as opposed to billets (short for billets de banque), which is paper money (cf. “bill”, “banknote”).