For example, the British in North America simply pushed back native Americans farther and farther from the east coast, while the Spanish destroyed the Aztec empire and enslaved the people. Why the difference?
Sorry, but the premise of the question isn't quite correct. Spain officially outlawed Native American slavery early on, while quite possibly the single most destructive demographic influence in Eastern North America was caused by slaving raids from the British, or those allied with the British.
After the excesses of cruelty seen during the conquest of the Caribbean, the Spanish Empire officially outlawed Native American slavery. Actual practice varied, Spain had a terrible time trying to control her agents abroad, but as early as 1512 the Laws of Burgos tried to limit the persistent exploitation of Native American populations. In 1542 the New Laws made yet another attempt to prevent exploitation, lead valiantly by Bartolomé de las Casas. The New Laws tried to gradually abolish the encomienda system, where tribute and labor was owed to the encomiendero in exchange for support and protection, but in practice gave way to situations that were no better than slavery. As previously stated, despite official bans against slavery actual practice varied. Rebels against the empire could be graciously enslaved (instead of killed), and we have evidence of ongoing slaving raids where political will or power failed to stop the practice. Nonetheless, slavery was outlawed in the Empire and would-be conquistadores, like Juan de Oñate, were forced to answer to the Crown for their cruelty against Native American populations.
As a contrast, the English colonies in North America began to exert influence along the Atlantic Coast a little late in the game of empires. Early fishermen, traders, and explorers routinely kidnapped unwary Native Americans from the eastern seaboard to sell into slavery, the most famous being Tisquamtum/Squanto who was kidnapped from the Massachusetts coast at least once, probably twice, before The Mayflower colonists appeared on the scene. Native American slavery in the Northeast persisted for more than a century, and the quest for cheap labor, in part, spurred the Pequot War, one of the first large-scale conflicts in New England that unfortunately set the tone for many later Native American-English interactions.
In the Southeast the effect of the slave trade was catastrophic. I'll quote from a previous answer about the Southeastern slaving raids for ease of reading...
Slavery existed in various forms in different Native American communities before contact. Throughout the Eastern Woodlands raiding for captives formed an integral aspect of warfare, and captive exchange was a crucial aspect of forging peace between communities. However, existing practices changed dramatically with the advent of the Indian slave trade.
Practices varied, but captives could be adopted into their new nation, rise to prominence, and their children could inherit a non-stigmitized status as a full member of the nation. Rushforth tends to present a less benign view of captivity in the Great Lakes region, where refusal to assimilate brought worse status and treatment, but those who adapted to their new state could experience a large degree of freedom.
In the Southeast Carolina slave traders transformed captives into goods that could be bought and sold. Instead of raiding neighbors in the cyclic pattern of small scale conflicts, larger parties set out on long distance raids with the sole intent to capture enemies for sale, not adoption. These raids were used with brutal effectiveness in Florida, where Carolina sought to undermine the Spanish mission system by enslaving their allied Native American population.
Accurate numbers will be hard to come by for this period, but here are a few best estimates from Gallay's work. Gallay believes 4,000 Florida Indians were captured and enslaved between 1704 and 1706. In 1708 the Governor of Florida, Francisco de Corcoles y Martinez estimated ten to twelve thousand Indians were taken from Florida. Father Joseph Bullones reported that four-fifths of the Christian Indians remaining in Florida after 1704 were killed or enslaved. The scale of raiding was so catastrophic that refugees fled south, hoping for transport and safe haven in Cuba. A ship captain carried 270 Florida refugees to Cuba in 1711, and said he left 2,000 Christian Indians and 6,000 more seeking baptism when he departed the Florida Keys. Gallay's very conservative estimate for the total number of people enslaved, not counting those who died in the associated warfare and displacement, in Florida alone is 15,000-20,000. The peninsula was practically depopulated of Indians by the early eighteenth century.
If we go further out to include the greater Southeast, Gallay's conservative estimates for numbers enslaved include 1,500 to 2,000 souls for the Choctaw during their coalescence, and 1,000-1,200 for the Tuscarora and their allies. Another few thousand from the petite nations along the Gulf Coast and the areas bordering French influence on the Mississippi. In the Piedmont 4,000-10,000 were enslaved. All told, his very conservative numbers suggest 30,000-50,000 Amerindians were captured directly by the British, or by allied Native Americans for sale to the British, and enslaved before 1715. Carolina exported more slaves than it imported before 1715. This number does not include those who died as a result of hostilities related to the slave trade, those displaced by the endemic warfare, or those who died as a result of infection and malnutrition common to refugee populations the world over.
So, as you can see, the English were quite happy to use slaving raids as a tool of war against Amerindians allied with the Spanish, with the added benefit of making money and displacing Native American populations from rich tidewater farmland. Unfortunately, slavery was an ever-present threat and tool of war in the Americas.
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