These terms seem to be used interchangeably.
Is there a difference between them for non-moving object on Earth, or moving objects for that matter?
The difference is that the centre of mass is the weighted average of location with respect to mass, whereas the centre of gravity is the weighted average of location with respect to mass times local $g$. If $g$ cannot be assumed constant over the whole of the body (perhaps because the body is very tall), they might (and generally will) have different values.
I don't see an immediate connection with movement though.
Centre of mass & gravity coincides until they have unifrom gravitational field. The time uniform gravitational field is lost we rather consider centre of mass than centre of gravity. However, they both're interchangeable.
Quoting from tha wiki page:
Center of gravity is the point in a body around which the resultant torque due to gravity forces vanish.
That means that for any rigid body, the two points are the same, because you can model rigid bodies in free fall as if gravity acted only on the center of mass, and forces on the center of mass make no torque.
Center of mass is actually the integral of mass density. Meanwhile, center of gravity is the integral of force of gravity!
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