The difference is that in one case, you have a privately owned and operating website exercising censorship as a function of the owner's property rights.
Just as you have the right to deny a stranger entry into your home, a website owner has the right to set the rules of engagement in their properties.
Whereas,
As in the case of the Australian Gov't, ostensibly based on common law, is exercising the bludgeon of government fiat in order to ban access for all citizens outright to a particular website, and in doing so, is engaging in censorship.
There is a difference between a gov't engaging in censorship reminiscent of Stalinist-era USSR and you telling a stranger they cannot enter your home without your permission.
The division between public and private is central to nearly all modern governments, even a "communist" gov't like China makes the distinction even though that gov't does exercise power at will, more or less.