What is the Difference between a Hash and MAC (Message Authentication code)?

The main difference is conceptual: while hashes are used to guarantee the integrity of data, a MAC guarantees integrity AND authentication.

This means that a hashcode is blindly generated from the message without any kind of external input: what you obtain is something that can be used to check if the message got any alteration during its travel.

A MAC instead uses a private key as the seed to the hash function it uses when generating the code: this should assure the receiver that, not only the message hasn't been modified, but also who sent it is what we were expecting: otherwise an attacker couldn't know the private key used to generate the code.

According to wikipedia you have that:

While MAC functions are similar to cryptographic hash functions, they possess different security requirements. To be considered secure, a MAC function must resist existential forgery under chosen-plaintext attacks. This means that even if an attacker has access to an oracle which possesses the secret key and generates MACs for messages of the attacker's choosing, the attacker cannot guess the MAC for other messages without performing infeasible amounts of computation.

Of course, although their similarities, they are implemented in a different way: usually a MAC generation algorithm is based upon a hash code generation algorithm with the extension that cares about using a private key.

A hash is a function that produces a digest from a message. A cryptographically secure hash is for which it is computationally infeasible to generate a message with a given digest. On its own a hash of a message gives no information about the sender of a given message. If you can securely communicate the hash of a message then it can be used to verify that a large message has been correctly received over an unsecured transport.

A message authentication code is a way of combining a shared secret key with the a message so that the recipient of the message can authenticate that the sender of the message has the shared secret key and the no-one who doesn't know the secret key could have sent or altered the message.

An HMAC is a hash-based message authentication code. Usually this involves applying a hash function one or more times to some sort of combination of the shared secret and the message. HMAC usually refers the the algorithm documented in RFC 2104 or FIPS-198.

A MAC does not encrypt the message so the message is in plain text. It does not reveal the secret key so a MAC can be sent across on open channel with out compromising the key.