I recently read that the mass that we deal with in the equation $F=M_{i}a$ is called the inertial mass and the mass that we deal with in $F=M_{g}g$ is called the gravitational mass.
I find a hard time understanding why we need these two different names. It's the stuff that makes up the ball that is responsible for pulling it towards the ground when dropped from a height and it is the same stuff responsible in determining how hard I should push it if I want to accelerate it on a level plane. So, why do we need to invent two different notions of mass in the first place? What is the fundamental difference between the gravitational mass and the inertial mass that one should not apriori ignore?
I have also read that the inertial mass and the gravitational mass are equivalent to each other and the general theory of relativity is based on this result. So, I feel that there must be physical meaning to the equality of these two masses--they shouldn't be trivially equal to each other. How can a fruitful physical theory (the general theory of relativity) be based on the equivalence between the two masses if they can be assumed equivalent apriori? In other words, I want to say that the equality between the two masses must be a physical result in order for a theory to stem out of it--it cannot be a trivial mathematical identity (otherwise, it would be physically unfruitful). So, what is it that makes these two distinct concepts necessary in the first place and what forces upon us their equivalence?