I'll preface this by saying that I'm not a medievalist, so I won't try to opine too much on the workings of medieval society. That being said, I think the key difference here isn't between a theist/non-theist basis for rule and law, it's a difference between a conception of a god that is transcendental (a la Christianity), and a conception of (a bunch of) ancient Gods that are immanent (i.e. dwelling within the world). The way you interact with a transcendental god who you primarily know about from religious/philosophical texts is fundamentally different from the way you interact with an immanent god that you primarily know about from ritual practice. In the former case, rules and laws come from attempting to carry out the will of the divine as epitomized by philosophy and scripture. In the latter case, rules and laws and decisions (including decisions like going to war!) are made in concert with gods through the practices of ritual and divination. Is this the only input into the way that societies worked in the ancient world? Of course not. But I would argue the same is true of medieval societies - not everything was motivated by religion either.
