That's an absolutely enormous question. The answer really depends on each particular region you're talking about. Also you're comparing former British Colonies in North America with former Spanish and Portuguese Colonies in South America; there is a large difference in the history of each place, a difference in culture, a difference in geography and resources, a difference in pre-colonial history, etc. There is also a significant influence of present-day politics and recent history such as American intervention in Central America that explains some of the implied questions in your post; look into the recent history of the region, and for example at the United Fruit Company as a specific example. Even as recently as 2007 the company (now Chiquita) pleaded guilty to funding a terrorist group in Colombia for protection of their assets. You could also look into the CIA-backed overthrow of the democratically elected government under Jacobo Árbenz replacing him with a Dictator in Guatemala in the 1950s. This led to a decades long civil war in which the - mostly US backed military - killed over 200,000 people. Also, many former colonies are doing quite well, Chile as an example, so I think your thesis is actually invalid. However, to go into a bit more detail briefly regarding North America and Jamaica, both being or containing former British Colonies:

After the French and Indian War, the French losses removed a significant check on British expansion (you can read more in Anderson's Crucible of War: The Seven Years' War and the Fate of Empire in British North America). With the expanding British power in North America, many native populations were dispersed and existing tensions between tribes were often exploited by the British to weaken any unified resistance to their colonization.

Keep in mind that in the USA and Canada, the settling peoples from France and Britain displaced and gradually killed the native population and kept pushing them further west; you could read, for example, Trail of Tears: The Rise and Fall of the Cherokee Nation, for a much later example that happened after the creation of the American nation. The point is that it often is not comparable to other colonies because the British did not really build a system in North America that integrated to any significant degree with existing peoples, they simply replaced them with Europeans working within a British system.

By the time 1778 came around and America declared independence, the states were comprised of mainly British, Dutch, and German people and were well connected in trade to Europe and each other, with growing economies, and well able to remain prosperous on the own, especially with French support.

While much of the economy of the southern states of course was dependent on the slave trade, North America as a whole was not. With the abolition of the slave trade in 1807, places like Canada didn't feel significant economic consequences. In the USA of course, slavery would continue for a while longer and ultimately be a component factor in the coming civil war, but even after the confederate loss, because the nation remained undivided, there was the chance for the country as a whole to continue to grow. The northern states did help with reconstruction, with both Lincoln and Johnson making efforts to bring the south back to "normal" as quickly as possible (see for example Donald's Civil War and Reconstruction).

This was not the case in places like Jamaica for example, which depended almost entirely on slavery, with the slaves outnumbering the rest of the population 20 to 1. Even after the abolition, many slaves continued to be smuggled in, and/or were replaced with indentured servants from China and India. Full emancipation didn't happen in Jamaica until 1838.

The population in 1834 for example was 371,070 with 15,000 white colonists, 5,000 freed black people, 40,000 of "mixed race" and a staggering 311,070 slaves (Donovan's Jamaica). This is not in any way comparable to the North American colonies in the overrepresentation of peoples in slavery.

After independence, Jamaica actually grew economically pretty well with a 6% growth rate for years. However, with significant inequality problems and poverty, along with growing corruption, they've had trouble continuing that which I think is largely a relic of slavery and the massive inequality it created. Combine that with being a small geographical region, an island country, and then look at the amount of natural resources available in North America combined with all the aforementioned history of independent economic growth and diversification; when viewed with these considerations in mind, it seems fairly straightforward. Again though, each case is different and many South American economies are doing well. A lot of the modern problems are caused by recent politics and history as much as more distant factors as well.