How is digital signature different from a message authentication code (MAC)?
The main functional difference is that anyone able to verify a Message Authentication Code is also able to forge one, because the same key is used for both tasks; whereas someone with the public key can verify a digital signature, but can't forge one.

Contrary to a MAC, digital signature is thus usable in contexts where the verifier is not trusted, which is of tremendous practical value. There's a price to pay for that: a digital signature is significantly larger, and slower to generate and/or verify, than a MAC is.