They can indeed be used as synonyms, but if a distinction is to be drawn then I would say that murmuring is more like whispering, in that the person doing it is difficult to understand because of the low volume of their voice. In contrast, a person who is mumbling is usually difficult to understand both because of the low volume of their voice as well as obstruction caused by the person neglecting to fully open their mouth and enunciate their words.
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