The original rumor that was circulating was that Shiny Pokemon in the show could evolve into other forms that normal ones couldn't. Also, there is another rumor that regular Pokemon can become Shinies, as sort of a semi-evolution. For instance, Ash's Pikachu may end up turning white.
And actually, in the show, shinies seem to be a bit stronger than the regular ones, but in the games, they really aren't, so... AAA!! This is confusing!!
I think Shinies are just Pokemon's version of Albinos. Due to the 'colorfullness' of Pokemon, they are different colors from just pure white.