Well, it would depend on where you are talking about in the southern hemisphere. In general though, because the southern hemisphere has a lot more water vs. land when compared to the northern hemisphere, there is not as sharp a contrast in temperatures because you cannot build up the arctic air masses like form in Siberia and northern Canada.
However, some pretty cold air builds up in Antarctica and can find it's way north into the lower latitudes and contribute to some pretty big storms.
Another factor to less snow in the southern hemisphere is the fact that a lot of the land areas are too close to the equator to even get cold in the first place.
The snowiest places in the southern hemisphere are all at higher altitudes like the Andes in South America, the mountains in southeast Australia and also in New Zealand.