I have been three times to Cuba; first time (in 1999) I went to visit a friend at the Health Ministry, and they had quite a good dialup access point; back then, dialup was still the main Internet access mode where I live (Mexico). The lacking part was, of course, computer access in the population.
The last time I was there (2010) was shortly before the connection to Venezuela started operation. I was invited to give a talk at the "Universidad de Ciencias Informáticas" campus, near La Habana. There, basically every student lives on-campus (the university is in a decomissioned Soviet base). All rooms have a computer — Old one, but working. And yes, network access was quite slow. Students also had a terribly low monthly bandwidth allowance (IIRC it was in the vicinity of 300MB), and after hitting that ceiling, there was no way to get more bits for them. It was quite interesting to see how a large group of people learnt to use the Internet with Javascript off, images off!
There was no censorship I could find (using a regular student account). Of course, I didn't go testing everything, as I didn't want to leave my host disconnected — But the main issue was the limits derived from having a single satellite uplink for the whole nation. I was told the situation improved vastly after the fiber to Venezuela was laid, but I cannot comment first-hand on it.
Of course, I'd expect now a fat fiber will be laid to Florida.