The Amazon add-on subscription is totally separate from a standard Acorn TV subscription and is exclusive to Prime members. All of the programming is the same, but you will not be able to sync your membership or watchlist across the two platforms or log in using the same credentials.
The Acorn TV Fire TV app (available in the Amazon app store) does NOT require a Prime subscription. If you're already an Acorn TV Member just download the app and log in using your existing credentials.