The main benefit of peppering is that if your password hashes are compromised (read: stolen) ... they can't be brute-forced. Password hashes such as argon2 make brute-force cracking more expensive, but nothing can help you all that much if the password is "letmein" or something quite high up on the guess list. With well implemented pepper there's just no way to even check one password, never mind brute-forcing.

If you go beyond peppering and also use encryption to protect the whole password database you can get additional benefits like keeping the list of usernames secret.