Do you only need to backup the 12 letter seed words and the private key, or both the public & private key?
Back up the 12 letter word, because that is how your whole wallet derives from.
If you only backup a private key then it could potentially mean that you lose out on coins that are sent to your other addresses - there is no means of getting your other addresses private keys with one of your private keys.
And you don't need to back up your public key, no. The public key can be derived from the private key at any given moment using sites like bitaddress. Run it offline if you want to have more security, but it's not necessary if it's just for an online wallet anyways.