It's key to note that not all North American colonies became rich after independence - many Caribbean islands, once some of the richest areas in the world, are now among the poorest places in the Western hemisphere. In evaluating outcomes of ex-colonies, colonial government structure is key. In places that were conducive to cash crop cultivation or raw material extraction, colonial governments were controlled by a small group, repressive, and unequal. In places that were not, colonial authorities were forced to develop human capital and internal markets to make colonies profitable. These styles of government persisted after independence: in the first category, local politicians simply took the place of colonial elites.
In the Western hemisphere, incomes generally decrease the closer to the equator one goes. This is not by accident - equatorial climates are more suited to the growing of cotton, tobacco, spices, and other crops that were in demand in Europe during the colonial era. Equatorial economies are not just poorer: they were historically less democratic. While Argentina, the Latin American country with territory furthest from the equator, introduced universal male suffrage in 1853, Brazil would not do so until 1891, Mexico until 1917, Bolivia until 1938, and Jamaica until 1944. In the 19th century, average level of education in the Western Hemisphere also decreased the closer one got to the equator. The conclusion one draws from these facts is simple: ruling a simple economy dependent on resource extraction, colonial administrators and post-colonial governments alike had no incentive to develop human capital. An educated population was a rebellious one, and since their plantations and mines did not require a plurality of educated workers, there was no point in creating a potential threat. Also without an educated population, post-colonial governments had no incentive to create a representative government. This perpetuated what Daron Acemoglu calls "extractive institutions": a system that benefits the personal fortunes of the ruling class and is disconnected from the interests of the people.
The economic model of colonies not suited to the production of exportable commodities was fundamentally different. In order to be exportable, a commodity had to be relatively scarce in the destination market and expensive enough per pound that it was worth transportation costs. New England and Canada, which had few of these resources, developed far more complicated economies. Their international trade activity centered around being part of the European manufacturing supply chain, which required increasing levels of education.
This distinction is consistent across post-colonial states worldwide. Those that were dependent on the export of a particular commodity have lower levels of education and lower average incomes than those which were not. Generally, the best off former colonies are those whose climates vaguely mirror those of their metropoles, examples being the US, Canada, Australia, Manchuria, Korea, and Taiwan - these areas were unlikely to be able to grow a crop that the metropole could not produce itself and were "allowed" to develop a significant industrial sector before decolonization. With some exceptions, Africa was relegated to a commodities extraction role across the board in the world economy. The well known phrase "Africa is resource rich" is largely meaningless: "Africa" is not one entity, resource distribution is inconsistent, and by what measurement? However, it is true in a sense: Africa is three times the size of Europe, had a vastly different climate than most of Europe (allowing the cultivation of different crops), and, due to historically lower population densities, the trans-Atlantic slave trade and tens of millions of deaths during colonization, was underpopulated. This underpopulation ensured that there was never enough labor to fully exploit the continent's natural resources or cultivate the acreage: across the continent, the surviving Africans living in easily accessible areas were corralled into mines and plantations. Those living further from the coast, rivers, and railroads were relegated to subsistence agriculture.
As in equatorial America, colonial bureaucracies, armies, and infrastructures were designed to keep a small elite in power, and this condition persisted in most of Africa after decolonization. The example of Congo is informative: the Belgian occupiers built railroads only to access resource deposits and plantations in the interior, and raised a de facto mercenary army called the Force Publique whose role was internal security. In trying to control one ethnic group, the Force Publique would deploy troops hailing from another. It should not have been a surprise that, following decolonization, the country quickly fell into the hands of an extractive autocrat. The fate of Congo is arguably the most extreme case in Africa, but similar situations can be found across the continent. Even in countries like Tanzania, where post-colonial authorities had a genuine desire to industrialize and change the economic model, decades of colonial rule had deprived Tanzania of the human capital necessary to make those industries profitable.
Sources:
Acemoglu, Daron; Robinson, James A. Why Nations Fail.
Acemoglu, Daron et al. Institutions as a fundamental cause of long-run growth.
Blanco, Luisa and Grier, Robin. Natural resource dependence and the accumulation of physical and human capital in Latin America.
Ochon, Moses. African Colonial Economies: Land, Labor, and Livelihoods.
Driscoll, Mark. Absolute Erotic, Absolute Grotesque: The Living, Dead, and Undead in Japanese Imperialism.
Settles, Joshua. The Impact of Colonialism on African Economic Development.
Okia, Opolot. Forced Labor and Colonial Development in Africa.