Cuba and Puerto Rico were actually given to the United States at the same time following the Spanish-American War in 1898. Unlike Puerto Rico, though, Cuba was granted independent status in 1902 (in a way; they had to agree to a lot of terms, see the Cuban-American Treaty of 1903). This was partially owing to the fact that Cuba had spent decades fighting against Spanish rule before being handed to the US and were not backing down from the fight. Due to its history of turbulent governments and economy, ultimately the US reasoned that giving Cuba independent status (and therefore, being left to control its own affairs, was no longer the US's problem) was more worthy of US time and resources than trying to keep it as a protectorate. See Louis A. Perez, "Cuba Between Empires," especially Chapter 14.

Follow-up question: Didn't Congress prohibit the annexation of Cuba?