What is the difference between verifying the hash value (MAC) and security provided by authenticated encryption? I read this post Difference between MAC and Authenticated Encryption? but I'm still confusing the difference.

Seems like MAC can guarantee both integrity and authenticity (tbh I dont know the difference of integrity and authenticity).

What is the difference of MAC and AE?
Is AE using MAC?
Authenticated encryption also keeps the message secret.

Often, though not necessarily, authenticated encryption is built out of a composition of a cipher and a MAC—which, if you have to do yourself out of parts like AES-CTR and HMAC-SHA256, you should do as encrypt-then-MAC.

P.S.  There is no substantive difference between authenticity and integrity.  They both answer the question: Was this message sent by the party I expected to send it?