The difference is small and pedantic:

“Once” means “A single time”

“Nonce” means “Used a single time”

Nonce carries extra information that something is being used.

Wikipedia claims this term dates to middle English:

Nonce is a word dating back to Middle English for something only used once or temporarily (often with the construction “for the nonce”). It descends from the construction “then anes” (“the one [purpose]”).

Nonce is used extensively in the context of cryptography to mean “used exactly one time”. Usually this refers to a random number that a server shares with a client to establish secure communications.