I noticed some "White Walkers" break into ice chips and others do not; some only fall down and are immobile. Why is that?
White Walkers "shatter" and Wights fall down.
White Walkers (also known as Others in the books) are inhuman beings with mystic powers. They are not dead but instead are a sort of magical humanoid life form. White Walkers have the ability to turn dead and decomposed bodies into wights. "Wight" is a catch-all term for the reanimated corpses of humans. Thus, only "some White Walkers" shatter into chips, because the ones that don't are not White Walkers.