Yes you're correct cinnamon and nutmeg do indeed come from the same tree.
Although the difference is that nutmeg is a seed of the tree while the cinnamon comes from the bark of the tree.
But they're pretty similar and come from the same tree but there is the difference between the two as in the one comes from the bark of the tree and the nutmeg is a seed.
So in reality when you're eating cinnamon you're eating the bark of a tree that has been ground up into fine powder and for the cinnamon sticks which are delicious they're just curled up cinnamon tree bark.
Just like you can eat bark from other trees as well but the cinnamon tree bark is more delicious.