I guess the best way to describe the difference between the Tablets of Bambur and the Code of Junil is that one is a more primitive "thou shalt not" while the other is "the right path is...". And I am horribly inarticulate to address such a difficult topic. If the moderators think this is flamebait, please remove as you please.
The RoK I'd imagine has a more ancient set of right and wrong that is a little removed from the standards we have today. In ancient times, morality was defined as duty to your community, staying in your place in society, upholding religious rituals, working hard... etc. ideas that help a somewhat honourable society survive. There's no concept of a "good person", only "good king", "good servent", and the concept of "good" means whether you have upheld your role or not. If you have not, then you were bad, even though it was from circumstances that was beyond your control. People would recognise that sometimes it was impossible for you to be a "good farmer", but you're still a "bad farmer".
Think of it as what morals meant in pre-Homeric times.
As to why RoK turns evil civs neutral... let's just say many of the commandments would outright BAN evil practices. What it does not do is attempt to explain why something is right and wrong.
Concepts like compassion, charity, redemption or justice are much more complicated and slow to develop. I guess to highlight what the RoK commandments do NOT encompass, I'll use our own world's concept of sacrifice and Sin. Sacrificing for gods to appease anger or prey for a good harvest has been in existance since forever, really. I would imagine Kilmorph will act much like one of the pagan gods, and her temples would routinely demand sacrifice to keep her favour (no little animals, of course!!). Then, in Christianity, our ultimate sin was paid for with an infinite sacrifice, both literal and symbolic. Maybe Kilmorph did accept an ultimate sacrifice from the dwarves, I don't know. But in Islam, however, they've moved even further, because they realise that while we should not be held guilty for the sins of others, even if we are in its karma-like effects, we cannot be "redeemed" by sacrificing others either. That's why they ask for forgiveness from Allah directly.
(of course, Junil is not The One, and that causes problems...)
And of course, whether you agree with the above statement or not, we've just entered a whole new level of morality.