The main difference that we can identify between these two stories is their purpose. The story "How the World Was Made” takes the form of a creation myth. It attempts to describe natural phenomena and explain it through the knowledge of the world that the Native people possess. On the other hand, "The Way to Rainy Mountain" is not a story that explains the world to the Native people, but one that discusses the culture, traditions and way of life of the Native American cultures.
This difference highlights the way that themes have changed throughout the centuries. While the stories were a necessary and basic part of daily life in past centuries, the stories nowadays mostly serve to preserve and value these disappearing cultures. These changes have most likely taken place due to the fact that these cultures have diminished and are struggling to remain alive.