Apologies for the confusion - unfortunately it is not possible to replace your display name with your Facebook name, ie the Facebook name does not override the display name.
If you'd like to have your Facebook name appear instead, you'll have to create a new account and then link it to Facebook using the instructions in this support article.
If you decide to do that, bear in mind you'll have to unlink your Facebook account from your old Spotify account (the one with the display name) before linking it to the new one.
You can also check out this handySpotify Answer on transferring your saved music and playlists to your new account.
Hope that clears things up! Any questions, let us know 🙂