The tie between Cuba and New Orleans was strong until the American embargo against Cuban which hurt the economy of New Orleans.
New Orleans was seen as the northern most Caribbean city and had strong economic ties to the entire Caribbean, including Cuba, throughout the 1700 and 1800s
Both Cuba and New Orleans had slaves and their slave cultures (and the culture of the free people of color) influenced each other.
In Cuba, authorities allowed slaves to meet in “cabildos”, groups of slaves of the same ethnic group. In those organizations they were allowed to practice their own religious and cultural rights (although they were all still baptized Catholic.) While the authorities might have meant this as a way to make sure the enslaved people did not unite across ethnic groups, it ultimately allowed them to build a stronger culture. In Cuba today, they still celebrate traditional ethnic festivities on January 6th.