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The difference between a hash and a digital signature is:
Hashes are used to verify the message integrity only, digital signatures are used to verify message authenticity and message integrity.
Hashes are used to verify message authenticity, digital signatures are used to verify message integrity.
Hashes are used to verify message integrity and authenticity, digital signatures are used to verify message authenticity only.
There is no functional difference
A hash is used to only verify the message integrity - if a message changes, the hash of a message will change, too. Hashing is not used to authenticate the sender. A digital signature is used to guarantee that a known source generated the message (non-repudiation), and that the message was not altered in transit (integrity). A digital signature typically includes a hash, subsuming hash functionality.
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