How does finishing the Trial of the Sword affect the Master Sword?

Completing each level of the Trial of the Sword will add 10 damage to the Master Sword (which starts out at 30).

You will notice that when fighting Calamity Ganon monsters with the Master Sword, it will start glowing blue and its damage will jump to 60 points (along with a nice bonus to durability).

Fully completing the Trial of the Sword will put the sword permanently in this state (glowing blue, 60 damage, 200 durability). There will no longer be an additional damage boost when fighting Calamity Ganon monsters, however.