Push is where domain is moved to a new account holder at the same registrar. Transfer (auth code) is where domain moves to an account at a different registrar
A push is a domain transfer between accounts at the same registrar. An auth transfer is from one registrar to another.
For example, you could push a domain from your GoDaddy account to another persons account at GoDaddy, but you would need the auth code to transfer a domain from GoDaddy to Namecheap.