A private key is the secret number that authorizes spending from a specific address. A seed phrase is a set of words that generates all of a wallet's private keys.
Every crypto address has a private key. Whoever holds the private key can sign transactions and move the funds at that address.
The seed phrase is the master secret. From it, a wallet can derive an unlimited number of private keys across many different cryptocurrencies.
You back up a wallet by writing down the seed phrase, not individual private keys. One phrase restores everything, which is why protecting it is so critical.
The private key controls one address; the seed phrase controls the whole wallet. Both must be kept secret, but the seed phrase is the more important backup.