In the context there is no real difference in meaning between the first and second example. There may be slight difference in emphasis, but either could be used and would be correct.
The last example is not very well written, it looks a little childish. When snow is falling we say "It is snowing". The perfect form would be "It has been snowing" (used when discussing a past action with effects in the present). In the final sentence, the speaker is describing a sequence of events in the past.
Snow started to fall -> I went to sleep while snow was still falling ->I woke up and got out of bed -> I saw that everything looked white.
(After a heavy snowfall, there will be a white layer of snow covering everything. See for example a news report heavy snow)
The use of the tense "It had been snowing" indicates that the start of snowfall had preceded the past action of going to sleep. However, this isn't a very well written sentence, and there are other ways of expressing the same idea.
When I went to sleep, it had already started to snow. I woke up the next morning and looked outside. Everything was white!
