Q:

What is the difference between center of mass and center of gravity?

What is the difference between center of mass and center of gravity?
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?

A:

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.

