Private key, public key, address: how they relate
Three things that get used interchangeably and should not be. What each one is, which you can share, and which one is the wallet.
These three terms describe a chain of derivation, each one computed from the last, and each safe to share to a different degree.
The private key
A very large random number. It is the only thing that can authorise spending, and it is never meant to leave your control.
Everything else about your wallet is calculated from it. It cannot be calculated back from anything else — that one-way property is the whole basis of the system.
The public key
Derived from the private key by elliptic curve multiplication. It can verify that a signature came from the matching private key without revealing that key.
You can share it safely. In practice you rarely handle it directly, because wallets show you the address instead.
The address
A shortened, checksummed form of the public key — hashed and encoded so it is easier to copy and so typos are detectable.
This is the only one of the three you are meant to hand out. Giving someone your address lets them send to you and nothing else.
Why the distinction matters in practice
A seed phrase is a human-readable way of storing the material that generates private keys, so it carries exactly the same weight as a private key. Treat the two identically.
If a site, an app or a person asks for anything other than your address, stop. Receiving money never requires more than an address, and anyone suggesting otherwise is either confused or working you.