What’s the difference between a mnemonic phrase, a recovery phrase, and seed phrase?

Nothing! They all refer to the same 12-24 sequences of words that grant you access to your wallet.

If you lose your mnemonic / recovery / seed phrase, you lose access to your wallet so it is very important that you keep a physical record of it in a safe place.

Best practices recommend you keep multiple physical copies and never keep a screenshot or digital record of it in case your device or cloud storage provider was ever compromised.