As I have found out, both 'Stottern' and 'Stammeln' refer to stuttering/stammering (correct me if I'm wrong), but are they being used in different contexts?
In German, these words are not exact synonyms.
Stottern is a problem in speech where you have difficulties in forming single words. It sounds like they don't come out of the mouth, the speaker repeats the f..f..f..first c..c..consonant multiple times before he can speake the word (or is forced to give up).
Stammeln rather means that the speaker cannot form meaningful sentences. This could include some Stottern, but would in general mean that someone brings out some words, or only parts of words, half sentences, repeated attempts to phrase something in different ways and so on, without really making sense.
The movie picture "The King's speech" is about a man who had big problems with Stottern, but apparently was very clear about what he wanted to say (if he only could), so he was not at all in danger of Stammeln.