What is the difference between the mnemonic and the private key? My thinking so far:

If I have a private number, then I can use that as the private key.

If I have a mnemonic, I can match up each word to a number in the wordlist to get the seed, which is the private key.

Is this correct?
Your mnemonic is creating an extending private key for deterministic wallets, which is different than a WIF Private Key. An extended private key can deterministicly generate millions of private keys, so that you can regenerate all those same addresses in the future as long as you have the extended private key or your mnemonic seed.

Example of extended private key:
xprv9s21ZrQH143K3S4Ab76xBk6gRfR2Z7Di2m1x75R8bLp6uETwpxB7A6KbbbEu4gPaEqSic6HCKBDAScGMpSHinsxJii7ao3EDWHDZixRp7g2

Here's a tool that can help show more of what's going on behind the scenes with a mnemonic https://iancoleman.io/bip39/