A hitter will generally grip tighter than a swinger because a hitter is 'fighting' rotation, and a swinger is 'allowing' it.
Imagine you were to swing a hockey stick like object, but with a very long blade. All things being equal, the blade will align itself with the direction/plane of force (why people slice, in part). A hitter resists the blade rotation and must provide a firmer grip to do so. A swinger, by contrast, needs to create the 'throw out' force (rotation) that will square up the blade. You could also argue that a hitter might tend to be more upright, and a swinger more flat because of those conditions, although there are no hard and fast rules there IMO.