Looking at the definitions for once and nonce. they appear very similar to me.
Under what circumstances would one or the other be a more appropriate word choice?
Once and nonce aren't similar enough to be interchangeable despite their both meaning "one time". They are also in different registers. Don't use nonce unless you want to sound pretentious. The most common uses of that word are in this snippet from Wikipedia:
Nonce word is a technical term, so it's not pretentious unless you know that your audience doesn't know it. For the nonce is easily replaceable with for the moment.