I am going to answer this by linking two other posts where I deal with related issues at length.
The short answer is that Cuba did not become independent during the initial wave of Hispanic American independence movements because of a combination of economic prosperity, contented elites which enjoyed a very beneficial collaborative relationship with the colonial government, and because wherever separatism did pop up, it was quickly and systematically suppressed by Captain Generals (basically Spanish military governors who ruled the island's political and military affairs) who were very good at their jobs. Elites, contented with the status quo and fearful of an uprising of the growing population of slaves, as had happened in Haiti, decided to cling close to Spain.
In the long term, Cuban separatists were divided over the question of slavery and the fear of an "Africanization" of Cuba and massacre of whites, again, thinking of Haiti. When separatism did spark rebellion in 1868, Spain was able to focus a large part of their military might in Cuba as opposed to all throughout Spanish America during the independence wars earlier in the 19th century. These rebellions were, in part, the result of the imposition of the 1837 Liberal Constitution in Spain which began the modernization of the colonial practices between Spain and its colonies, to the detriment of Cuban economic and political interests (think along the lines of colonial American anger over taxation without representation). In the late 19th century, after the destruction of the rebellion after a long ten year struggle (1868-1878), Spain attempted several key reforms in Cuba but these proved to be too little and too late. In 1895 the second independence war broke out and was eventually turned into part of the broader Spanish-American War (1898) in which the US defeated Spain and basically removed what was left of that kingdom's American and Asian Empires.
Let me know if you have follow up questions.