I think, there's little difference, when using ganz or ziemlich. Ganz is a little bit stronger than ziemlich, but there isn't much space in between meaning. Most time the difference is vague and indecisive. I think most people will use them spontaneously equally without seeing any difference, though they know/feel a difference subconsciously.
In the end, it's like comparing very and quite in English. Very is more intense and similar to extremely. Quite add strength as well but is less powerful. Compare very[=extremely] heavy snowfall vs quite heavy snowfall. The tricky thing is that ganz can also share the meaning with pretty, fairly as in pretty good or fairly difficult [=to some extent but not very].
First things first:
Mir geht es ganz schlecht.
Mir geht es ziemlich schlecht.
This first sentence means, that I am feeling very sick. I am not able to do anything and I just want to stay in bed.
In the latter I am also in a weak constitution, i.e. not healthy. I am feeling sick, but I am able to do the most necessary work (I can go to the grocery store, but though I wish someone else would do).
Du bist ein ganz starker Gegner.
Du bist ein ziemlich starker Gegner.
Using ganz means your opponent is very strong and you need much endurance, concentration to beat him. You have to go a high pace, much deceptions, ... Saying sehr starker Gegner, however, sounds much better.
Using ziemlich still means your opponent is good but you are better. It's not easy to beat him, though. You still have to fight, but you can allow yourself doing mistakes and forgive good chances without risking your win.
Regarding your example (but I will turn to the opposite, as it's less common to talk of ganz leichter Schneefall):
Es ist ganz (schön) starker Schneefall.
Es ist ziemlich starker Schneefall.
Aside first: Using also schön sounds better to me. I don't know if it just regional, but in this example I would never omit it. In this case it makes the meaning even more powerful, more intense like very very much. Again, it sounds slightly better to me using sehr starker Schneelfall.
Now what I want to express by saying the first sentence:
Let's imagine the situation that I am driving my car through a heavy snowfall. Then I mean: 'I don't see much'. I have to drive very slowly and carefully. I have to concentrate very much on keeping the lane.
Same situation for the latter one. In that case, I focus on the fact that the snowfall is disturbing, annoying. I still have to drive carefully, but I drive a lot faster because it isn't that dangerous.
Both variation can also stress my astonishment how much snow falls, i.e. that the degree of intensity is more than I expected.
As mentioned above, ganz can also mean the pretty same thing as pretty does:
Das Essen war ganz gut.
In this context, ganz can be translated to pretty, or fairly. It wasn't good but not bad either. It was okay.