Cuba remained Communist after 1991 for much the same reasons that Vietnam, Laos, and China did--the Cuban Communist Party had developed a political/security infrastructure independent of the Soviet Union. None of the Warsaw Pact states (Romania, Poland, East Germany, etc.) ever experienced an organic grassroots Marxist revolution like Cuba did--those countries were simply occupied by the Soviets as a consequence of the Third Reich's massive defeat in the Eastern Front of WWII. When Gorbachev's government declined to quell rebellions in Eastern Bloc countrues, their Communist infrastructure simply collapsed (as would have been the case in East Germany in 1953, Hungary in 1956, Czechoslovakia in 1968, or Poland in 1981, had the Soviet government/military not intervened directly in those uprisings).

Cuba never had the same dynamic with the Soviet Union as did the Warsaw Pact states, in part because of its huge geographic distance, but also because the Cuban Missile Crisis was essentially the peak of Cuban-Soviet relations and interdependence. Remember that Che Guevara began denouncing Soviet "revisionism" shortly after the Missile Crisis dissipated, and he was still officially lauded by the Cuban government for decades afterward. The Soviet Union was Cuba's largest trading partner (and Cuba did undergo a significant recession in the 1990s), but by the late 20th Century the Cuban regime had achieved sufficient autarky (and authoritarian control) to weather the storms of 1989-1991. Legalizing non-communist political parties was the death knell for the old Soviet system, but Castro always insisted that allowing pluralistic democracy would leave Cuba vulnerable to regime change, so he never allowed it. Chavez's Venezuela also became a great ally and trading partner for Cuba just in time to cement the Castro regime's survival in a post-Soviet world.

As to why the U.S. failed to influence (or invade) Cuba "given how close the island is to the U.S. geographically," I suspect it might have something to do with the role the Bay of Pigs plays in the Cuban and American consciousness. No American president wants to repeat Kennedy's great embarassment, and the odds of overthrowing the Cuban government without great bloodshed in 1991 were even slimmer than they'd been in 1961 (when the Cuban Revolution was still quite new and inchoate). On the other hand, Cubans (like the Vietnamese) were energized by the fact that they were able to repel an attempted American invasion (even if it was just a CIA operation), and resisting American values is considered more patriotic in Cuba than it would've been in, say, Communist Romania (a place that the U.S. government never even considered invading). Thus, the U.S. was wary of intervening in Cuba, and many Cubans were eager to prevent intervention, so no Grenada/Panamesque regime change occurred.