Protection and its counterpart share a secret
Symmetric encryption uses the same secret for encryption and decryption, unlike the public/private division. The users may be communicating parties or one entity protecting stored data; two people need not be online together.
Only authorized participants should obtain the key. Public algorithms are compatible with this model, but placing the key where everyone reading the ciphertext can also read it defeats the intended secret-key protection. Concealing an algorithm name is not key management.
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Encryption is only one service
Secret-key mechanisms also authenticate messages, derive keys or wrap other keys. Their input, output and security goals differ: a MAC does not conceal contents, and encryption alone does not automatically authenticate them.
Authenticated encryption supplies a defined combination of confidentiality and integrity. Applicable nonce, length and usage rules still matter; choosing a block-cipher name without a multi-block mode and verification behavior is not a complete communication scheme.
Combining with public-key mechanisms and separating uses
Hybrid systems often establish keying material with asymmetric mechanisms and protect bulk application data symmetrically. This division neither makes symmetric cryptography obsolete nor removes responsibility for protecting shared secrets.
Directions, phases and purposes need protocol-defined key separation. NIST guidance generally limits a key to one purpose to reduce cross-mechanism interactions and compromise impact; designed combinations such as authenticated encryption have their own rules rather than licensing arbitrary reuse.