While you're waiting for a response, I'd recommend reading this answer from /u/yodatsracist in a thread asking, "Why did North America split into 3 countries and South America split into 15 though they were colonized similarly?" It contains some comparisons you may find informative.

It's also worth considering why you perceive North America to be more prosperous. North America is comprised of 23 countries, some of which endure extraordinary poverty (Honduras is one of the poorest countries in the world -- comparable to Haiti -- and yet it is part of North America), while South America is comprised of only 12. North America does not stop at the United States' southwestern border, or even at Mexico's southern border. While the continent contains the most prosperous nation in Earth (the United States), it is more difficult to compare wealth on the continental scale.

Latin America (which is what I think you mean by "South America") is vast, nobody can give a good answer to explain all its economical history. We can split Latin America in three regions that are more homogeneous: Spanish continental America, Caribbean and Brazil. These regions had very different political and economical evolution, and I am only qualified to give a brief answer on Brazil. So I will shift your question to: what are the main differences that explain the discrepancy in economical development between US and Brazil?

Both countries are, at first glance, pretty similar. Both were "discovered" three years apart, have similar sizes, both had indigenous populations whose system of government was dispersed and tribal (comparing with the Aztecs or Incas), climate in southern US and southern Brazil are similar and both were colonized by Europeans. What could explain the difference, then? I will split my answer in three parts: Colonization, Independence and Industrialization, which I think are the main differences between the economical history of these two countries.

Colonization

Brazil was occupied and exploited by the Portuguese since its discovery, which was 120 years before top-hat men landed on Plymouth rock. Portuguese model was clear: extract valuable resources and sell to Europe to capitalize on the heavy investment of the great navigations. Sending boats to the ocean was expensive, sending an expedition to Brazil was expensive, the Portuguese empire wanted returns on the investment and any notion of "country building" was never part of the process. The very first product to be heavily extracted and exported from Brazil was a rare kind of deep red wood, named Pau-Brasil, which gives the country's name. By the end of the 1500's Pau-Brasil was nearly extinct, Portuguese interests turned to sugar, which will be Brazil's driving economical force until being replaced by coffee.

Compare this with the Anglo-Saxon model of settlement of the Americas. Portuguese (and Spanish) interests were not nation building, but rather exploration and exploitation. This will define all colonial policies, and will also define who is being sent to these regions. These were not well educated pilgrims who were fleeing religious persecution, but farm workers sent to explore and exploit a tropical land, and, of course, a frightening number of African slaves. Settlement and nation building was a consequence rather than a goal for Portugal. This mentality was ingrained in the Brazilian ethos, specially because the nation was made of mostly slaves and descendants of Portuguese farm workers whose trip to Brazil was almost a form of punishment.

Independence

Inspired by the US, in the end of the XVIII century we will see pockets of independence movements in Brazil, most of them organized by landed elites who wanted to pay less taxes. These movement would not lead to independence, though. When Napoleon invaded Portugal in 1808, the Portuguese royal family promoted Brazil to "adjoint kingdom of Portugal" and fled to the colony when threatened by the French. After the fall of Napoleon, the royal family decided to go back to Portugal and re-establish Brazil as colony, but it was a little too late. In 1822, Pedro I, the son of the Portuguese king João VI, declared Brazilian independence and became Emperor of Brazil. Contrast this with the American revolution, try to imagine the prince of England declaring the US independent and becoming king. Not only that, Pedro I (as he is known in Brazil) would eventually abdicate the throne when João VI dies and go back to Portugal to become Pedro IV, leaving his infant son to rule Brazil as Pedro II.

What you must understand from Brazilian independence in the context of your question is: it was a peaceful process that barely rustled any feathers of the elites in power. This is also one of the reasons why Brazil is so large and did not split in 20 different countries. The status quo was preserved and our economy just kept rolling with the same old rules imposed by the Portuguese, but now profits would stay in Brazil. This meant a double down in cultivation of few products (coffee, cotton and sugar, mostly, there is even a coffee leaf in the Brazilian Imperial flag), total domination of politics by the landed agricultural elite, enforcement of slavery until the very end (only abolished in 1888, one of the last countries to do so) and heavy resistance to adapt to the changing in the economical model of the world, which brings us to the next point.

Industrialization

This is a big one. While the US had its industrialization forced upon it with the 1812 war, Brazil never had such a struggle. When the Monarchy ended in 1889 (yes, the slavery question was one of the knives that killed the empire), the Republic was established and organized by the agricultural oligarchies that had ran the country since forever. Our first president was a military leader that was forced to depose his friend, the emperor. The second president, Floriano Peixoto, repressed revolts so brutally that he once massacred more than 160 citizens of a city in revolt and changed the city's name to honor him to further humiliate it (have you ever been to Florianopolis?). Third president, Prudente de Morais, was the first civilian on the job and had to put down way too many fires. To appease the clash between the many agricultural factions, regions and states, the fourth president, Campos Sales, will create the "Great Deal" that will define Brazilian politics until 1930: the federal government will always support the gubernatorial candidate of the current ruling party in any state, and all governors will support for president the candidate that the current president appoints.

These candidates all depended on the support of the agricultural elites due to the voting system. Vote was not secret in Brazil, and you would not want to vote against the interest of your boss. More than 70% of the Brazilian population was working on plantations at the time, so having the agricultural elite's vote was the only way of doing politics. This system would force a president, Rodrigues Alves, to sign the devastating "Acordo de Taubaté". I mentioned this "Treaty of Taubaté" in another post here , it basically forced the federal government to buy agricultural debt from landowners, and bankrupted the Brazilian state in less than 20 years.

You can now grasp the level of rigidity of the political system and how attached it was to agricultural elites, since pretty much the independence of Brazil. One of its by-products is the total lack of stimulus to industrialization, and effectively a suppression of industrialization efforts. Brazil was 100 years behind the US in industry, and the US won big time when Europe was ravaged by the first world war by supplying it with industrialized goods. Brazil chose time and time again to double down on the commodity economy and paid a heavy price. As my American history teacher would say: "It is not completely absurd to compare Brazil to what the US would be if the south had won the civil war".

Conclusion

This entire post is an over-simplification of 400 years of economical history, but I believe these are the seeds of the answer you want. Colonization with nation-building purposes, disruptive independence, early industrialization and a difference in export strategy were the main differences in the economical history of these countries and the XX century rewarded greatly a geographically isolated and industrialized country whose population was more literate and well educated.

Sources:

Boris Fausto, História do Brasil

Edgard Carone, A República Velha (instituições e Classes Sociais)

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