When your mage unlock a Master Perk, It also gains an ability reflecting that skill tree: 

Alteration Master - Grants the Mystical Attunement ability.
Mystical Attunement: Increase the duration of all spells by 25%


Conjuration Master - Grants the Empowered Soul ability.
Empowered Soul: Increase Magicka by 50 points.


Destruction Master - Grants the Elemental Mastery ability.
Elemental Mastery: Increase damage of Fire, Frost and Shock by 25%.


Illusion Master - Grants the Phantom Menace ability.
Phantom Menace: Summon Jar Jar Binks, the world Destroyer. NO, THAT'S NOT IT: Increase all spells magnitude and duration by 50% when used against unsuspecting targets. (unsuspecting = not in combat)


Restoration Master - Grants the Indomitable Spirit ability.
Indomitable Spirit: Increase health regeneration by 50%. Effect is doubled when health is below 50%.




So as you can see, the more proeficient your mage is in the schools of magic, the stronger he will get. The abilities are very useful in the school tree they are from, but can increase other elements as well, making the quest to become a pure mage a very rewarding one. 

If you want more strong things nerfed and weak things buffed, please check this: 
SKYRIM FOR VETERANS - https://www.nexusmods.com/skyrim/mods/85011/?