Puzzle pieces are regular puzzles you can earn as a reward for completing various activities in the game and winning levels. They are automatically placed on the puzzle set when you enter it for the current month, but if unused Puzzle pieces are left by the end of a month, they do not carry over to the next month.
Rainbow Pieces are collectible pieces which you can earn in the Lucky Balloon game mode. Once 10 Rainbow Pieces are collected, you can exchange them for a single Rainbow Puzzle Piece.
TIP: You can exchange Rainbow Puzzle Pieces by entering any Puzzle Set on an album. If you enter a currently active set, tap the Rainbow puzzle at the bottom left of the menu and then select Trade. If you enter any of the previous Puzzle sets, tap the green trade button!
Rainbow Puzzles can be used to complete literally any Puzzle set in your album. Tap the rainbow puzzle and select use to place it manually on any highlighted spot on the set.
TIP: You can check the number of owned Rainbow Pieces and Rainbow Puzzles while in the album on the bottom of the game screen.