Power Gems adds new materials to Minecraft allowing for more powerful suits of armor and weapons. Each gem adds special abilities to the armor as well. The Aquamarine Gem gives you a boost in jumping for example:

Tourmaline aids in regeneration so you won’t need a food source while wearing the armor. So as soon as you take damage your hearts will start to refill even if your food meter is depleting.

Lapis now has a much more functional purpose of aiding in water breathing.

And Citrine gives you fire resistance and its accompanying sword causes fire damage.