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
With the private key, I can import funds from a wallet, for example, Electrum to Bitcoin Core without the need for a prior transfer, just to concentrate UTXOs in Bitcoin Core and send the funds there.
HD wallets that support seed phrases don't allow key import, and if they do, they do so in a crappy way.