Like the Dream World Plushies, the puzzle pieces have no further
use in this game except to function as collector's item only and
for users to have fun finding and collecting them.
Completing a puzzle doesn’t do anything else, like Umbreonpanda
said, but the puzzle pieces themselves do serve the function of
giving you more Exp. Shares if you collect enough of them.
ETA: You get one Exp. Share after collecting 10 puzzle pieces,
another at 50 puzzle pieces, another at 150 puzzle pieces, another
at 300 puzzle pieces, and another at 500 puzzle pieces.