Let's go back to the beginning and consider what might happen if the sugar beet were commercialized a century or so early. For purposes of the discussion, suppose a high sugar strain was developed around 1700 rather than close to 1800 and that over the next 50 years the problems of refining and of finding proper crop rotations are solved, such that by 1750 large scale cultivation and sugar production is going on.
Beet sugar is not going to out compete cane sugar on a purely economic basis, especially not when the cane is produced by cheap slave labor. However countries without colonies will certainly encourage its production on mercantilist principles to keep positive trade balances. This was the reason for Prussia's interest OTL and we can expect it to spread through Germany and into Poland and Russia, probably other European countries too. The larger producers will forbid or tax imported sugar, restricting the carrying trade in Caribbean sugar from the maritime powers. The added supply will also generally push down prices for cane sugar, similarly to the events of the 19th century OTL. Sugar islands will no longer be the goldmine that they were in the 18th century.
This might change the lead-up the Anglo-French side of the Seven Years War, if their Caribbean islands are seen as less vital. Maybe not though, since the immediate causes of the war lay in North America. In general there might be less action in that region in the War of the American Revolution and the Wars of the French Revolution too.
The slave trade will continue, but the rapid development of Haiti's plantation economy might not occur. Or it might, since coffee was also a major crop there. The same will go for the development of Cuba as a slave plantation economy at the end of the century. Brazil would be heavily affected too.
I'm sure there's more to consider, these are just a few thoughts.