• The inertial mass m_i determines how the body accelerates as a results of the application of any force.
• The gravitational mass m_g determines how the body "feels" a gravitational force (and how much of a gravitational force it generates).

The fact that one can equate the above two forces:

mi A = G mg M R-2

Then if m_i equals m_g then one (correctly) sees that the acceleration (due to the force of gravity) is independent of mass.

This fact

• is a mathematical confirmation of Galileo's experiments (cannonball & feather etc)
• can be used to derive Kepler's 3^rd law of planetary motion

En route to deriving Kepler's 3^rd law Newton appears to have over-looked (ignored) the fact that (at that time) there was no explanation why m_i should equal m_g.

The fact that that simply says m_i and m_g are indeed equal is often referred to as

Galileo's Principle of Equivalence

An alternative way of stating it is that inertial and gravitational forces are the same phenomenon.

This principle can of course be tested and potentially proven incorrect (& thus General Relativity disproven too). However, currently this tests have indeed shown m_i and m_g are equal to the limits of tests (1 part in 10¹¹).