Do 饶舌 and 说唱 mean the same thing in Chinese?
饶舌 and 说唱 do have the same meaning in Chinese when you are talking about music.
Taiwan hiphop singers translate rap to 饶舌.
Mainland ones translate rap to 说唱.
As an example:

mc_hotdog (from Taiwan) uses 饶舌 to refer to rap in his lyrics.

kindergarten killer（from the mainland）uses 说唱 to refer to rap in his lyrics.


The relation between 饶舌（Taiwan) and 说唱（Mainland）is just like the one between graduate (U.S.) and postgraduate (U.K.).
But when you are not just talking about music, 饶舌 have a wider definition, just as Raynoceros said.