Message integrity
Message integrity is provide via Hash function. The hashing function is used are 1 way Hash function which means given a data it will produce a unique hash for it..
Receiver on getting the message+sign ,calculate the hash of the message using the same 1 way hashing function once used by the sender. Then it compare the computed hash with hash it obtained from the decrypted digital signature. If both hash matches , it means that data is not changed during the transmission or message is same as what sender has sent.
Message Authentication
Message authentication is provide by the use of asymmetric keys.
Sender sign the message digest using it private key and send this as digital signature along with the message.On receiving , Receiver will then try to decrypt this message digest (signature ) using the public key of the sender. If it can decrypt the message digest using the sender public key , then it can know for sure that the message was sent by the sender only as he only can have his private key that was once used to encrypt it .
Now your doubt,
in diagrams you find in book for either of these will have same parts, but what is important to note here is that the message hashing is for checking integrity check and encrypting or decrypting the message digest is for authentication .
Hope i cleared your doubt.