This is an immensely complicated question, and I don't think it's possible to really answer the question in a single comment. That said, I would first like to recommend some books that I think are really useful to understanding the problems of underdevelopment in sub-Saharan Africa.
The African Colonial State in Comparative Perspective, Crawford Young
Markets and States in Tropical Africa: The Political Basis of Agricultural Policies: Robert Bates
African Economies and the Politics of Permanent Crisis, 1979-1999 , Nicolas Van De Walle
One thing to keep in mind is that Africa is not a country. There is a tremendous diversity in the political structures, economic bases, and historical influences on the countries of Africa. I am going to focus on Tanzania and compare it to India in this answer, and what I write here only sort of applies to other nations. Moreover, I want to emphasize that my answer is a selective narrative that both simplifies and ignores data points, because trying to create a coherent narrative out of such a complex series of events is really hard.
The first feature I want to highlight is that European colonizers were able to largely sidestep native elites in Tanzania. Zanzibar in the 18th century was at the heart of a complex commercial network that spanned the entirety of southern Africa. However, the late 18th and 19th centuries was the heyday of slavery. Demand in the Arab world, and on plantations on Seychelles and Mauritius. A series of states emerged whose prosperity depended upon access to arms to hunt and capture slaves. It's a system that allowed some people to get wealthy, but was ultimately destructive to trade, agriculture, and manufacturing. An example is the Nyamwezi, centered around Tabora, in NW Tanzania. They were the dominant group in the slave caravan trade in the region, but their economic power and influence declined dramatically after German conquest. German and later British colonizers were able to rule over Tanzania, with little need to partner with native Africans.
The British who arrived in India were just as greedy as those who entered Africa. However, India already had a powerful commercial elite, that the British could not work around. The British East India Company was always poorly capitalized, and Indian merchant groups played an important role financing colonialism. For example, local bankers financed the British bribing certain factions to change sides at the last minute at the Battle of Plassey, the battle where the British established supremacy in Bengal. This class would grow and expand in partnership with the British, and at inedpenedence India had a class of merchants and industrialists capable of challenging the British both in India and abroad. The Mumbai textile industry was outcompeting England in much of Asia, and Tata was building steel mills as early as 1907.
British rule over Tanzania was fundementally oppressed. The local chieftancy was forced to collect regressive poll and hut taxes from poor farmers. A small indigenous middle class of clerks and teachers emerged, but it was given little real political power. The British had this interventionist idea that ordinary farmers were incompetent, so they tried to force them to farm in modern ways. For example, the Tangayika groundnut scheme in NW Tanzania the British tried to force the Sukuma farmers to completely change how they farmed. I don't want to get too much into the agronomy, but the British massively misunderstood the soil, climate etc. The whole scheme fell apart, and several farmers got shot after they refused to continue to work for such poorly concieved schemes.
I don't have exact numbers off the top of my head, but India was ruled by a shockingly small number. There were only a thousand or so administrators for all of British India, and the British relied extensively on Indians even at the highest levels. Something like 40% of all positions in the Indian Administrative Service, the elite of the elite bureaucrats in British India were Indian by independence. Moreover, the British actually helped Indian farmers gain economic self sufficiency. They built the largest network of irrigation to keep Punjabi farmers prosperous because they relied on the Punjabi middle peasantry for their armed forces, and dramatically reduced taxes and rents in much of southern and western India. When India gained independence, it had a trained core civil service, and both farmers and merchants (who bankrolled the independence movement) with real political power.
The first president of Tanzania, Julius Nyerere, was truly an impressive figure. Personally honest, and he had a vision of a united and modern Tanzania. Yet he struggled to put this into practice. There were few Tanzanians in senior civil service positions, and the new government inadvertantly picked up many of the same ideological beliefs as the British. Nyerere was committed Socialist, and aimed to transform Tanzanian agriculture by collectivizing the small peasant farms, into large capital intensive modern farms. He planned on using North Korean and Soviet technical advisers, and like the British, ignored ordinary farmers. These collectivized farms were highly inefficient, and barely functioned.
In a lot of ways, Nehru, India's first Prime Minister was like Nyerere. Both were honest, visionary, socialists. But Nehru had a cadre of highly competent civil servants, so the schemes he initiated were much better run. But perhaps more importantly, ordinary farmers had real political power. India was a democracy, but more than that the ruling Congress was dependent (like the British before them) on the middle peasantry and rural elite. Instead of collectivization, the government was pressured into investing massively in irrigation, new seed varieties, and fertilizer. India saw a green revolution as food production increased dramatically.
I chose Tanzania as my comparator country because unlike many other African countries, Tanzania had relatively honest leadership, almost no ethnic conflict, and geographical access to the sea. But complex factors of political economy continously encouraged leaders to make bad decisions. I also chose Tanzania because I want to emphasize this isn't a permanent situation. Tanzania has been growing by about 7% a year for about the last 15 years. This is beyond the scope of the sub, but sub Saharan Africa is in it's longest period of sustained periods of growth, and there is good reason for real optimism about Africa's long term economic future.
There are a number of hypotheses:
The geographic hypothesis, advanced by books like Guns, Germs and Steel is that geography is a main determinant of development and prosperity. Very briefly, places with harsh climates or too much heat/wet do poorly, while places with mild climates and easy ground for agriculture and convenient waterways and ports do well. Most modern scholars think this hypothesis is lacking as anything more than a partial explanation, but it's important to know that it exists.
Another potential explanation is that culture is a major driver of development and prosperity. This argument is (at least partially) advanced by David Landes in The Wealth and Poverty of Nations. The idea is that cultural attitudes can spur development - the classic example being the 'Protestant Work Ethic' in Northern/Western Europe. Even Landes, however, concedes that culture can really only be a portion of the explanation.
The hypothesis I find most convincing is the institutional hypothesis advanced by economists Daron Acemoglu and James Robinson in Why Nations Fail. Peruvian economist Hernando de Soto is another advocate of this view point, particularly his book The Mystery of Capital where he focuses on the value of secure property rights and institutional access for the poor. The institutional hypothesis says that what kind of institutions a region has determines long run growth (and thus prosperity) - specifically whether they have inclusive institutions or extractive institutions.
Inclusive institutions are institutions that
Extractive institutions are basically the opposite of that - institutions that
The question of why some countries are rich and some are poor is on some level a question of 'Why did the Industrial Revolution happen where it did, and spread where it did?' I think Acemoglu and Robinson make a convincing argument that the Industrial Revolution happened specifically in England and the countries surrounding it because the institutions there were some of the most inclusive in the world at the time - encouraging private investment, invention, exploration, entrepreneurship, etc. Those things needed a specific institutional framework in order to truly flourish, and England's institutions at the time provided it.
Translating that into the more recent past, after England why did some countries Industrialize quickly and some are still struggling to do so? Again, I think the institutional hypothesis has great explanatory power. When Europe used their head start in industrialization to colonize Africa, they put into place brutally extractive institutions. Political power was almost always concentrated into the hands of an elite minority. Rule of law was shaky at best in many parts/most parts of the continent. Governments frequently abused the rights of their citizens in horrendous ways. The colonizers set up extractive institutions to govern their colonies, and the transition away from colonialism in many/most cases did little to improve those institutions. Often power was simply handed from an outsider elite to a local elite, who proceeded to extract from their countries nearly as badly as the colonizers had.
Where you do find African success stories, they are almost always where the institutions lean more towards inclusivity - countries that have working democracies, or where the rule of law is far more established and trustworthy, where legal systems protect individual rights reliably, etc.
This is a very brief overview of a very large question - why are some places rich and some places poor. Multiple books could be (and have been!) written about this question without getting to the bottom of everything. Many factors play a role in the end - certainly geography matters at least a little bit, culture matters at least a little bit, institutions definitely matter, and probably many others as well. Further reading that I would suggest:
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