Completing the game will let you begin a new game in Hero Mode, by over-writing your completed game save.
In Hero Mode, hearts will deplete twice as fast, and hearts and heart flowers will not appear unless you are carrying a heart medal.
Additionally, you will begin the game with a fully powered up Skyward Strike attack, and will keep all of your collected insects and treasures from before.
The Sheikah Stone will allow you full access to all of the hints since you will have already completed the game once.
Finally, the boss rush challenge will also include the final boss as a foe.