Question

Is there a difference between the center of mass and the center of gravity, of an object?

Answer 1

Yes there is, but in most practical applications, this can be neglected.

Say you have a homogenous sphere (same material through and through). Then the center of mass will be exactly at the center of the sphere, as this represents all the small particles with mass.

The center of gravity is dependent on what attracts the sphere, and its distance. For a (relatively) small sphere on earth the difference in distance between the lower and upper particles can be neglected.

But take the Moon.
Its average distance is 384000 km, and its diameter is 3500 km.
The distance between the "front" and the "back" is in a ratio of `1:1.009`. Since gravity falls off by the square of the distance, the front part gets attracted allmost 2% more than the back. And if we integrate all little gravities into one, we will have to conclude:

The center of gravity of the moon lies a little bit closer to us than the mass-center.

(The same goes the other way around, that's why we have tides)