Do 饶舌 and 说唱 mean the same thing in Chinese?
Is it about spoken vs. written?
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.
饶舌, a kind of popular music, (for me) it usually means difficult to understand what the singer sings/speaks, because they sing/speak too FAST to understand or too FAST to follow the lyrics.
說唱 not only for popular music, but a kind of performing arts also, such as 相聲(crosstalk). 相聲是一種說唱藝術(Crosstalk is an art of talking and singing)
饶舌 說唱 usually have the identical meaning in daily life.
It sounds to me that 饶舌 is a type of 说唱.
饶舌 is more like something hard to speak like tongue twisters. The lyrics or words are twisted(绕在一起) and hard to pronounce and singers are skillful.
说唱 is more general for any types of speaking forms in singing. No matter the lyrics or words are hard to pronounce or not.
If you can read Chinese, this addresses it well:
说唱就是纯说没什么语言上的技巧，饶舌就是玩技巧，玩文字游戏。 说唱指曲艺表演，RAP既为说唱乐。说唱乐在世界各地广泛传播。 饶舌，有唠叨、多嘴之意。饶舌歌是黑人英语发展而来，黑人英语融合了黑人民族原有英文的普遍性，经由黑人饶舌歌手快歌快语的说唱，饶舌歌曲不但用字简化许多，在文法使用上，也有所转变，自成一格，更与白人英语形成极大分野。