The last piece of the Special Demo that has been referenced to a few times already is the Pokemon Link. Once you have your Pokemon Sun or Pokemon Moon Game (released November 18, 2016), you can connect your Special Demo to it to transfer all the goodies you obtained within it.
Simply insert your game (not necessary if you have a digital copy of the game) but go into the Special Demo first. Go to the Professor in the Pokemon Center and select the option to send your items to the Pokemon Link of your full game. After this process has been completed, access the Pokemon Link on the main menu of your game and receive all your items and Pokemon.
The following Pokemon and items can be obtained in the Special Demo and sent to your Full Game through the Pokemon Link:
Ash-Greninja - Obtained at the start of the Demo
Nugget - Obtained by completing the Capture Challenge on Ten Carat Hill
Star Piece - Obtained by completing the Battle Challenge on Ten Carat Hill
Stardust - Obtained by clearing all the rocks on Mahalo Trail
Pretty Wings (x10) - Obtained by talking to the man outside the Pokemon Center one day after starting the Demo