Daniel B. answered 11/03/21
A retired computer professional to teach math, physics
The two concepts are the same for objects with uniform density, but not otherwise.
For example, consider two spheres of identical size.
Their geometric mean is half way between them. And that is true independently of their density.
In contrast, their center of gravity will not be half way if one sphere has larger mass than the other.
Note that the difference between "center of gravity" and "geometric mean" depends on the density, not their shape.
An object may be very irregular in its shape, yet have its center of gravity at its geometric mean as long as its density is uniform.