Both of them are onomatopoetic words--that is, the word is an imitation of the sound it is describing. "Mumble" means indistinct and hard to understand. In general a "murmur" is a smooth, pleasant, gentle noise. The sound of tree leaves when the wind blows through them is a "murmur." In the case of speech, "murmur" suggests a smooth, quiet, even pattern of speech. You might say lovers "murmur sweet things to each other." You would not say they "mumble sweet things to each other!"

Mumble is to talk in a way that people can’t understand. Often because it is too quiet or indistinct. Murmur is more talking quietly, but clearly, as a sort of background noise. It seems an unusual word to use. “Whispered” might be better.