Well, идти has a lot of different use cases in Russian, such as (in no particular order):
- functioning (usually about watches) - "часы идут"
- run through - "дорога идёт через лес"
- broadcasted by TV, like in "этот сериал идёт по НТВ"
- used for, applied, like in "эти доски идут на забор"
- to suit, like in "эта шляпа ей очень идёт"
- about rain and snow or related atmospheric events - this is the case you've discovered
- to take intentional actions in particular direction, like in "мы идём на эти непопулярные меры"
and even more.
You actually can say "падает снег" but it's more literal - it's about the actual fact that snow is, well, falling, while "снег идёт" or "дождь идёт" are exact Russian counterparts of English "it's snowing" or "it's raining".
Also, there's a verb "дождить" and I must say that the phrase quoted in the wiktionary "На земле дождила пасмурная хмарь" is extremely, outrageously bookish and honestly nobody talks like this in real life, but don't be put off by this, this verb is no stranger to casual conversation. One can look outside the window and say: "Дождит".
There's no similar verb like "снежить" or something though :)
UPD: As @Quassnoi correctly pointed out, "снежить" is something that actually exist (though de-facto used more rare rather than "дождить")