So, the basic answer is that the Italians, French, Spanish, Portuguese, and Romanians do speak Latin, it's just changed a bit.

While there is a distinction even in the "Classical" period between high and low Latin, the evolution of Latin into the romance languages only begins to occur after ca. 550, when the old Roman educational system begins to fall apart and intellectual activity moves into the monastic sphere. This shift in Latin was universal, and did not distinguish between class boundaries. It is this late Latin which became the romance languages, not "vulgar Latin".

The general awareness that there is a difference between Latin and proto-romance languages occurs ca. 800 in Francia and ca. 950 in Italy, at which point there are reform efforts. Indeed, it is the Carolingian reforms of the late 8th and early 9th century that created "Latin" as something distinct from what was being spoken, effectively creating a dead language.

There are no decent books on this in English, but if you can read French, see:

• Riché, Pierre. Education et culture dans l’Occident barbare, VIe-VIIIe siècle. 4e éd., rev. et corr. Points H195. Paris: Editions du Seuil, 1995.

• Norberg, Dag Ludvig. Manuel pratique de latin médiéval. Paris: A. et J. Picard, 1968.