The Atlantic slave trade involved the large-scale deportation of West African slaves to sugar plantations on the other side of the Atlantic.
Why was it more profitable to do that, rather than to grow sugar in West Africa?
West Africa has about the right latitude and rainfall for sugar production.
In short, Brazil and Caribbean Isles were easy to colonize and suited to the culture of the sugar cane.
This related question will provide most of the explanation why Africa was harder to colonize, and less welcoming to Europeans.
Moreover, it is far easier to control slaves outside of Africa. They can't hide in the local population.
Also, sugar became more common and sugar beets were discovered and used. West Africa was mainly colonized by France, which was the first world productor of sugar beets. So the mainland had little interest in promoting it.
When the Europeans showed up in the New World, they brought disease that killed a large portion of the local people (60%? 80%? 90%? ). This meant, from the European standpoint, that in the 1500's and 1600's, New World land was available for the taking.
West Africa, on the other hand, had a high population density. Europeans were militarily not strong enough in the 1700's even to take and control African land; it would have been too expensive to defeat the Africans in war to take the land, and the Europeans would have had difficulty in holding the land as well. It just wasn't feasible to steal African land.
Why didn't the Africans grow sugar and export it to Europe? Cash crops and subsistence farming are a different enterprise; with its higher population, subsistence farming was widespread in Africa, so once again, there was not enough land available to do cash cropping.
Edit: Mayo aptly points out that Europeans had issues with Tropical Africans diseases, especially Malaria. This is an excellent point, although yellow fever killed plenty of Europeans in the New World.
The cultivation of sugar was more profitable on islands. The plantations in the "Caribbean" were on islands. Until the mid-16th century at least, production from "Brazil," largely came from Santa Catarina Island (or coastal strips on the mainland with the properties discussed below). On the other hand, "West Africa" had relatively few islands except for e.g. the Canary Islands, and seaports on the "continent" were not generally convenient to the best growing lands. There are several reasons why this is important:
First, sugar is heavy relative to its value. It is hard and costly to transport, especially by land. The location of plantations on islands (or very near the coast) meant that transport distances overland to a a seacoast were relatively small. Second, and related, sugar needed to be "milled" before moving, and importing these mills was much easier on an island. Third, it was much harder for slaves to escape from an island than from a mainland. This was doubly true if they looked "different" from free people, as was the case in the New World, than would have been the case in and around Africa. Finally, it is much easier to defend an island against hostile "natives" than it would be to defend a plantation located on say, the West African mainland.
Edit:
As two commenters pointed out, it's true that sugar in Brazil was also produced on coastal strips rather than islands, but the real issue was easy "access to the sea," which the west African sugar-growing areas often did not have. There was a "Gold Coast" and an Ivory Coast" but no "Sugar Coast."
Each of the current answers is excellent, but is only a partial answer. Another piece of the answer that I'm surprised hasn't already been mentioned is the Atlantic triangle slave trade: "Molasses to rum to slaves."
While it is certainly true that the Carribean is well-suited for the cultivation of sugar and that it lends itself to slave-based plantations, due to the difficulty of slaves escaping and fading into the local population, every product needs a market, or there's no profit to be made.
In the "molasses to rum to slaves" triangle, the market for Carribean sugar was in the Americas (specifically, New England); it was less expensive to ship it to New England from the Carribean.
Additionally, there was demand in Africa for goods from New England, specifically rum (produced from Carribean molasses) and textiles. Without each of these legs in place, there would have been less profit in the others.
A strictly African model of sugar production may well have been profitable. However, with the addition of these two additional legs it became obscenely profitable.
Update:
@guenthmonstr made an excellent point in the comments that I wanted to address, but was unable to do within the character limit of the comment field, so I'm updating my answer to address it.
The comment was:
Triangle slave trade is a result, not a cause. A triangle need only develop where there are 3 geographically dispersed factors of production. In this case: plantation friendly land, slave labor, and manufacturing. If plantation friendly land had been available in Africa, then Africa might have simply exported molasses or rum to Europe or New England in exchange for manufactured goods.
My response:
I agree with everything you say, with the following qualifications:
You state that the "triangle slave trade is a result, not a cause." You are correct that this trade triangle was the result of the relative location of the three products. But the OP's question was about why sugar plantations in the West Indies were more profitable than those in Africa, and it is a contributing factor to that profitability.
The profitability of a trade triangle is as much about the location of the products relative to one another as the products themselves. If either the product or location is wrong, there's little or no profit. In this case, both were ideal to maximize profit. Again, it's a result of the location, but a contributing factor to profits.
You are absolutely correct that a system of African sugar plantations might well have been profitable, and that the African demand for textiles and other goods might have been satisfied from Europe. However, by this point in history, the British Empire was dominating production of textiles, and the colonies in New England were dominating that production, due to their close proximity to the raw materials. Transshipment of these products through Europe would have added to their cost, impacting the overall profitability of the trade. Trade would still have been profitable, but probably not as profitable.
Historically, when it can be made to work, a triangle trade always has a greater profit than two-way trade, due to the fact that the initial investment at any point is multiplied not once, but multiple times before returning its yield to the investor.
