The basic difference is that Turner belongs to an older historical tradition that sees westward expansion as something of a story of progress. It sees white civilization meeting Indian savagery and gradually overcoming it. In the process, the whites transformed themselves and their country.
By contrast, White's sees the frontier as a set of relationships and interactions between whites and Indians. He sees it not as a march towards a triumphal end, but a set of relationships that occurred in a specific time and place. He also looks much more at the meaning of the frontier for the Native Americans. He tries to see them not simply as savages (nor as people who were simply victims of white aggressors) but as real people trying to adjust to the new world in which they found themselves.
In this way, White's view of the frontier is a more well-rounded one that puts less stress on the glory of white settlement and more on the whole picture of how that settlement impacted everyone it touched.