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.
@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.
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.