Think of the centre-of-mass as a "weighed" geometric centre. It is just the entire object's mass "averaged down" to one point: $$r_\text{CoM}=\frac{\sum rm}{\sum m}$$
The centre-of-gravity on the other hand is the "averaged down" pull from gravity. If gravity pulls equally in each particle, then the total pull "averages out" to the "weighed" geometric centre, so it is the same as the centre-of-mass.
But imagine that gravity of some reason only pulls in the bottom half and not the top half. Then the total pull doesn't "average out" to the centre of this object - instead it "averages out" to some point in the bottom half. So, it does not coincide with the centre-of-mass in this case. Whenever gravity is non-uniform throughout the object, this may be the case (unless the symmetry of this non-uniform gravitational pull happens to make them coincide again).