Cuba is actually one of the reasons that the USA maintained diplomatic relations with the USSR during the Cold War. It was only following the Cuban Missile Crisis that a direct-line between the Kremlin and the White House was set up, so that the two parties could easily communicate in times of crisis.

The main reason that the USA shut down all relations with Cuba, was because Castro nationalised all private businesses in Cuba, a large number of which were American. America had close relations with Batista's government, and many American businesses were heavily invested in Cuba. This then led to the American embargo, and was only exacerbated by Nixon declaring that Castro was a communist and a puppet of Moscow.

Ultimately, it comes down to the fact that the USA needed to maintain reasonable relations with the USSR, and it did not with Cuba. The embargo on Cuba was symbolic, and made a strong statement that communist nationalisation of american business, especially this close to home, was not acceptable. Having nuclear weapons that near to America was also not acceptable. The USA did not want another Cuban Missile Crisis, and so they developed diplomatic relations with them. However, USA-USSR trading peaked during the period of detente under Brezhnev.