Why do we need a separate center of gravity idea when we have CoM?

[Ryder Rude]

Isn't that what I have?

Sorry I was viewing it on mobile. I had to slide it to see it full

 

[Dale]

The center of mass point literally follows the path that a particle of mass M would've followed if the forces on the system were thought of as directly acting on that point.

If the center of gravity (or buoyancy or any other distributed force) is different from the center of mass then the force acts through the center of gravity (etc) and therefore produces a torque about the center of mass. The motion of the body then cannot be described simply by the path of the center of mass. The object will rotate about the center of mass. Modeling this torque is the reason for distinguishing the center of mass from the center of gravity (etc).