Where did Snow go?

(Early-to-mid game spoilers ahead)

In Final Fantasy XIII-2,

in Sunleth Waterscape after you kill Mutantomato, Snow disappears.

Why does this happen, and why don't Serah and Noel experience something similar?

Snow is

a paradox. He has a L'Cie brand, which means he's on a Fal'cie quest. While the nature of the quest isn't stated, at this point one might guess his quest is to save Fang and Vanille in the crystal pillar from the flan or to find Lightning to make Serah happy again. He was sent through a time portal to 300 AF. Once the flan paradox is solved, Snow wouldn't go on that quest and so he is reverted. That Snow no longer exists.

The scene is shown in this video at around 4:30

why don't we dissapear too

Because, we can see and resolve paradoxes without being affected (which has happened many times in the story before this point).

Super advanced hyper spoiler Do not read:

Much much later in the game, you can come back and defeat His Royal Ripeness (the still paradox'd version of Mutantomato). What happens to Snow then?