How did Europeans use Christianity to justify their conquest of North America and its people? What were some differences in the approaches of the...

I would argue that European interest in the New World had much less to do with creating a religious utopia and more to do with interests in exploration and material exploitation. 

For the French and the Spanish, Christianity was merely the tool that was used to justify conquest. Before race was invented and used as a tool for conquest, Europeans created the division of Christian vs. heathen, or savage. Native Americans, because they had not been baptized as Christians, were the savages. Their absence of Christian faith and Christian manners provided, in the minds of their conquerors, the justification for their exploitation, enslavement, and massacre.

For the English, things were a bit different. By the mid-16th century, the Anglican Church had been formed by Henry VIII. His daughter, Elizabeth I, would rule as a Protestant queen and spent much of her reign fending off infiltration by Catholics, including her cousin, Mary, Queen of Scots. 

The Protestant Reformation resulted in the creation of numerous new Christian sects, though Catholicism remained firmly implanted in France and certainly in Spain. In England, the Pilgrim and Puritan sects were established. They inevitably faced persecution. Their response was to leave England. 

William Bradford was a Separatist leader, and later a signatory of the Mayflower Compact. He had grown up in Yorkshire, England. For a time, he lived in Leiden, Holland with his fellow Pilgrims, believing that, perhaps they could find refuge there. 

His vision, however, expanded into not only into seeking refuge for his sect, but also in creating a religious paradise in the New World. When he and his fellow Pilgrims established Plymouth Colony, they set up an agrarian community, in which they all shared the fruits of their labor. As the governor of Plymouth Colony, Bradford set up a government that separated religious concerns from secular ones. This premise stands in sharp contrast to the kind of government that was established by the Puritans in Massachusetts Bay Colony. The Puritans began arriving in the New World shortly after the Pilgrims. The Pilgrim community also tolerated dissent, whereas the Puritans did not. 

I'm summarizing a lot of background here to help you understand that the French and the Spanish had a very different purpose compared to the early English settlers. The French and the Spanish entered the New World looking for material gain. It is important to note that the French established a very lucrative fur trade in Quebec and parts of Northern New England. They merely used Christianity as an excuse to justify their exploitation of territory and their massacre of natives.

The essayist Michel de Montaigne wrote a short essay on this subject in 1580. The English title is "On Cannibals." I would recommend reading this to give you some additional context. Montaigne uses the treatment of the natives by Europeans as a lens through which he views his society. He writes about the way in which Christianity has been used as justification for murder and exploitation by the French.