You can get snow in Australia at a surprisingly northern latitude (we’re talking sub-tropical Queensland here), but it’s a rare event. The Snowy Mountains are just the bottom part of the Great Dividing Range, which runs up the eastern part of the country, all the way to top, in the tropics, and with unusually cold weather the snow will fall on these mountains much further north than usually. It snows a few times a year in the towns in the Blue Mountains, an hour west of Sydney (part of the same range). This is usually a light dusting - just enough to get a few seconds of footage of kids playing in it for the evening news. Sometimes there are more significant falls there. One year in the early 1970s, the town of Lithgow, north west of Sydney, had a white Christmas (of sorts). When it snows in the Blue Mountains, Sydney tends to get a bitterly cold wind, which we’re getting at the moment, so there may have been some falls up there. The western suburbs of Sydney are also prone to winter frosts.
It has even snowed (lightly) in the suburbs of Sydney, but I don’t remember ever seeing it.
We ain’t all about sun, surf, and sand. Much of the country can be downright bleak and miserable in the winter. We don’t mess with any of that -20 degrees shit though. Or the shovelling your driveway sort of stuff.
And yes, as has been mentioned, Australians will use the word ‘mountains’ rather more readily than will Europeans, North Americans, or Kiwis. We do have mountains, and they are beautiful and impressive in their own way - just don’t come here expecting alps.