Why did Newton introduced inertial and gravitational mass in the first place? This might be a silly question but I really don't understand what was the reason for which Newton ended up differentiating mass into inertial and gravitational. Why did he think it necessary to do so?
For example, an object is made to accelerate over an almost frictionless surface and then the same object is dropped from a certain height. The same objects in both the cases, the same amount of matter in both the cases, but they are said to be different.
The mass in first case is said to be inertial mass and that in second case is called gravitational mass. But why did newton think of classifying mass? And also what is the significance of this differentiation?
 A: Newton differentiated between gravitational and and inertial mass because it is a priori not at all self-evident that the inertial mass, as used in his second law, should be the same as the gravitational mass, as used in his gravitational law. It could be that a pound (measured with a scale) of brass has a different inertial mass than a pound of iron. The general equivalence (proportionality) of inertial and gravitational mass is actually highly surprising. Newton tested this equivalence (proportionality) experimentally by investigating pendulums with bobs of different materials: gold, silver, lead, glass, sand, common salt, wood, water, and wheat, and found inertial and gravitational mass to be equivalent within 1%. See, e.g.,  this link to the pertinent contents of his "Principia".
A: It is not so much their differentiation, as it is their identity, that is extraordinary. 
Consider the following. Inertial mass as defined describes an object's resistance to forces that attempt to change its motion. Described in Newton's second law, 

F = d/dt(momentum) = m * dv/dt

for a mass that does not change with time.
The inertial mass can therefore be seen as the scaling factor between the (net) force acting on an object, and the resulting change in motion as effected by the force. 
The gravitational mass, on the other hand, describes the magnitude that a certain object is affected by a gravitational field. Looking at it in either the form 

F = GMm/(r^2) = gm

where G is the gravitational constant, and M is the mass of the other (large) mass. In that sense then, if we were to consider g as being a quantity intrinsic to the gravitational field (rather than the object), the gravitational mass m then becomes a scaling factor between the strength of the gravitational field, and the force experienced by an object due to the influence of the gravitational field. 
There is little theoretical work (and none legitimate) during Newton's time that can explain why these two scaling factors should be the same. Considering both as intrinsic properties of the object, why should the tendency of an object to be affected by gravitational fields, be related in any way to the resistance of the object to changes in its motion due to external forces? The equivalence of the two values (equivalence principle) is therefore something that needs to be experimentally verified, and the search for a difference between the two has been ongoing since his time. 
As an aside, it is perhaps important to note that other forces of attraction and repulsion (as understood at that time) all have their own unique scaling factors. Be it the charge of particles for electrostatic forces, or the magnetisation strength of certain materials for magnetic forces, it does not seem like any correlation exists between them or with the mass property of the object. The uniqueness of these two values stands out.
Taking a closer look at the two formulas:

F = m * dv/dt    | F = m * (GM/(r^2))

The greater the resistance of an object is to being moved, the stronger is the attractive force exerted by an object on neighbouring objects to pull them towards it. Now, consider a ball placed on a taut bedsheet. The deeper the ball sinks into the bedsheet, the harder it is to move the ball. But, with a deeper depression created by the ball, the tendency for neighbouring balls affected by the depression to roll towards the main ball increases. This is a really, really crude way to think about general relativity, but I suppose it could pass off as a tiny bit of inspiration. 
A: *

*Inertia resists acceleration.


*

*Mass resists linear acceleration, $\sum F=ma$.

*Moment-of-inertia resists rotational acceleration, $\sum\tau =I\alpha$.

*...



Mass is just one type of inertia.
Now, on a completely different note, several different forces exist. Friction, drag, normal force, electric repulsion/attraction etc. And weight. They all depend on something. Drag on speed, friction on the normal force etc. And weight just happens to depend on mass.
It also depends on other things; distance for example. But also mass. Consider it a coincidence, because in classical mechanics there is really no better reason.
