The private key is a very long letter/number. But do most people write that down as well?
Nope, if you have the seed phrase there is no more need to keep a copy of your private key. Your private key, after all, is derived from your seed phrase. Having the seed phrase under your sole control means you also have the sole control of your private key. If in the event that you specifically need your private key for whatever reason, you could always extract it from your seed phrase using iancoleman's BIP39 tool.
What if you only have the private key but not the seed phrase?
What specifically about it? Take note that your seed phrase could give you your private key but not the other way around. I mean, you cannot derive your seed phrase from your private key.
So if what you have is a private key, you could import it to a wallet and be able to take control of your funds, but only those that are in the address whose private key you are in possession of. Generally, 1 address = 1 private key, but 1 seed phrase = unlimited private keys. So if you have a wallet and lost the seed phrase but have the private key of one of the addresses, you couldn't possibly retrieve those funds that are deposited in addresses other than the one of which you have the private key.