A crucial point should be made before answering: There was a broad diversity in worldviews among both Native Americans, who included people from the Arctic to Patagonia; and Europeans, who came from a plethora of different places and social classes. So it is hard to speak of "Native Americans" or "Europeans" without first including this caveat.
Often, though, differences in worldview had to do with different conceptions of land ownership. Europeans tended to believe that land could be held by individuals, to be passed down to their progeny. They also thought that land not "improved," that is, farmed according to European standards, was fair game to be taken. Indians, on the other hand, tended to think of land ownership in communal terms. It is categorically not the case that they had no concept of land ownership, but rather that they did not view land as a source of value with an objective price that could be sold to others.
Another difference had to do with religion. Natives associated spiritual with political and economic power. Europeans did as well, to an extent, but did not grasp that when Indian people adopted some aspects of Christianity, they did so to add to their own spiritual/political power. They incorporated it into their value systems and made sense of it in their own terms. Europeans, with the exceptions of some missionaries, regarded this as heresy.
Finally, it is worth talking about differing notions of government and politics. Europeans tended to assume that political leaders they dealt with spoke for all of their people (sometimes because it was convenient for Europeans to see it that way.) In short, many Europeans were incapable of comprehending the many factions that comprised many Indian policies. Equally, they did not tend to recognize when certain Indian peoples played one group of Europeans off of the other, trading with both the French, for example, and the British.