I have no clue. Ask Newell, Cauron, or Yates. I like to assume that he dyed his hair. I suppose you can do the same thing.
Actually, both vesions of Flitwick are played by the same actor. The version with the brown hair and short moustache was originally just supposed to be an unnamed wizard, a choir director and nothing more, but for whatever reasons, they decided to call that wizard Flitwick and keep that appearance. Rowling prefers the first, older version, with the white hair and large whispy beard, as its closer to what she imagined Flitwick to look like, and it better resembles his distant part-goblin ancestry.