Many Native American myths present a pantheistic view of the world, seeing God as being in or a part of all things. In many of the myths, Nature itself plays a role in creation. In some, we see the "earth-diver" motif, where some kind of creature dives down to the bottom of a large body of water to bring up a particle of earth on which other lands will be created, or to bring up fresh water so that humans and animals can continue living on earth. In some of the Coyote myths, Coyote helps to create various parts of nature (ie. stars) or even man - where Coyote uses his cunning to outmaneuver the other animals in their ideas of what man should be life. Other creatures, especially many of the trickster heroes, also play a role in creation, especially in bringing fire or the sun to mankind.
Because of these kinds of myths, Native Americans feel a sense of brotherhood with all creatures, thus leading to a sense of respect in how they treat animals and the earth. While they are willing to kill animals for food, many rituals, ceremonies, and prayers of gratitude are often associated with this, and it was typically their practice to use as much of the animal as possible to honor its sacrifice.
Many of the early American explorers, by contrast, saw the earth as something to be conquered and subdued - something with riches to be discovered and taken. The belief in "manifest destiny" saw all of America's lands as there for the taking - to be used as people saw fit. Though Native Americans might already be living there, that didn't seem to matter, for since they hadn't built up the land, they were not considered to be putting it to good use or to have any real ownership of it.
Additionally, animals were there to provide not only meat, but also sport. As a result, killing animals just for the sake of killing - for the fun of it - was perfectly acceptable to early settlers and explorers. Buffaloes could be killed just for their tongues, otters just for their fur, whales for the oil they might provide, etc. The whole carcass might be left behind to rot, but that was no reason for a sense of guilt. That was the way of the world - the earth, in these early explorers' minds, had been given to them by God to rule over and subdue, and that was their plan.
Early Native Americans believed that nature was the beginning of life and that all life eventually returned to the Earth. Similar to the notion of the "Circle of Life" their beliefs gave them hope of resurrection of the soul into another physical body that would continue on within this circle. Life to the Native Americans continued on as the inner soul of an individual was eternal in its existence.
Whereas, New World explorers gained their views/beliefs from the European ways of Christian beliefs. Their views/beliefs focused on the idea that one's spirit returned to a "heavenly" realm, rather than circling back to nature or continuing to dwell upon the Earth. The explorers (for the most part) did not believe that the soul would live forever upon the Earth, but would return to live with a "God from on high".