How did the life of a slave on a Caribbean sugar plantation differ from a slave on a Southern US cotton plantation?

In many ways not a lot, in other ways quite a bit. It all depends on specifics. Namely, which plantations and which year.

The advent of sugar plantations was incredibly early in the colonial era, in fact they were one of the first American colonial enterprises which had exports back to Europe. Mainly under the Portuguese and the Spanish. For many years in fact the British way of acquiring sugar was to raid Spanish ships. One Spanish spy even claimed they were so successful that buying sugar in England was cheaper than buying it in Spain or even the Caribbean itself. In later years the British and the French took the smaller Caribbean islands. The Iberian sugar plantations didn't last tremendously long and these smaller islands took over.

One particular aspect that was quite different was attitudes towards feeding slaves. The French law about this was the most filmsy out of all the colonial laws, so how slaves were fed differed greatly. Some allowed their slaves small plots of land to plant in on weekends. Some gave their slaves food directly (though in French colonies this appears to be less common). But quite a lot paid them in rum (given its links with sugar), which they were then told to trade for food. Many just drank the rum and starved. When food was distributed it didn't differ too greatly from slaves in other places in terms of nutritional impact. US slaves in the south got cornbread and pork, Carribean slaves got fish and cornmeal/rice, in Brazil they got manioc flour and dried meat. Some historians argue this was ample, if monotonous food but they were almost certainly ignoring the rum exchange traditions and focusing exclusively on the rationing of proper foodstuffs.

But in many ways the treatment of slaves would have been quite similar, these laws also applied to mainland colonies (when they had them). What differed greatly is how cotton compares to sugar. Cotton can be stored and worked quite some time away, sugar not so much. The sugar industry relies heavily on the connection between the processing and the growing sectors, to such an extent that most plantations processed sugar in house. This link is so fundamental rather than splitting heritage (as was common) sugar plantations and their processing units were typically given only to the best son as a package. This means that the average Carribean sugar slave had not only to learn to harvest the cane, but also to process it, to boil it, to shape it etc. This is not a small task, sugar was one of the worlds most labour intensive crops until the 20th century, especially in the early colonial era when they used 11th century production techniques!

Sources: Sweetness and Power: The Place of Sugar in Modern History, Sidney Mintz 1985

The Feeding of the Slave Population in the United States, The Caribbean, and Brazil: Some Remarks on the State of the Art. Ramiro Guzman, 2013