Much as you say one can easily interchange them, Big Ogg with the club wants to remind you about the law of the jungle part of "natural rights".
It's the concept that rights aren't given by God, but that we, simply by virtue of being human, have certain rights that can't be taken away.
By virtue of being human? Why? Big Ogg needs to feed his hungry pet lion & you run slower than the deer. Even you can see the logic in feeding a hungry lion so it won't starve. What gives your life greater value than the life of Ogg's pet lion?
Natural rights are derived from natural law, which in general encompasses both the concept of "god-given rights" and "natural rights."
If you wanna be more than lion food, ya gotta give me more than "natural law".
And Cardinal Richelieu thinks you're in the way of his power so you are just going to have to die. If you're a Protestant and lucky, maybe he just stripped you of your political rights and protections (what you probably consider now to be God-given rights).
Natural rights, god-given rights, what they actually are is always determined by people.