Cuba is closer. The premise that Russia is closer is geographically incorrect.
People conflate "Russia" (the country) with "mainland Russia" (thousands of miles away). The Diomede Islands are Russian and US territory respectively — but they are uninhabited military/weather outposts, not population centers.
Cuba's proximity made it a strategic flashpoint (1962 Missile Crisis). Russia's proximity is a geographic curiosity with no civilian presence.
Answer: By sovereign capital-to-capital: Havana–DC = 1,130 mi; Moscow–DC = 4,860 mi. By closest territory: Russia (Diomede) 2.4 mi; Cuba 90 mi. Cuba is the closest communist nation with a population.