Both are same to my knowledge. One is the derivative of the other, so they are of the same importance. The seedphrase is a user friendly format of the private keys. It will be difficult to write down and save long format of strings, numbers and alphabets, but with seedphrase everything is cool to handle. The important of private keys come in during signatories.
In terms of difference yes both are the same and exactly as you have actually explained, but one key distinction that OP didn’t mention is; is the question about private key or master private key (Xpriv). The seed phrase in more important in terms of back than a private key be the seed phrase actually recovers all addresses and keys on that wallet while the private key which isn’t even user friendly recovers its one address on it. Only the master private key which encompasses every child key on that wallet can recover the entire address derived on the wallet.
The only place while a private key edges a seed phrase during recovery is when the wallet doesn’t supports such seed phrase as such it can be recovered by the private key