Nothing game-breaking. But, when the
SAME Pokemon picks up 5 Shiny Leaves, you go talk to your friend, and they'll turn it into a Shiny Crown and place that on said Pokemon. This also gets you a new rank for your Trainer Card, so you're one step closer to having that fully completed.
The chances of finding a Shiny Leaf decrease with each one you get, and the locations are dependent on your Pokemon's nature.
Bulbapedia has a good page detailing Shiny Leaves, so that'll probably help with where to find them.