They are similar, but there is a difference. "To mumble" or "mumbling" is like when someone talks in a way where their words are not really coming out of their mouth. Imagine a drunk man talking to himself. Or a shy person who is too afraid to speak loudly to someone. A "murmur" is more like when you can hear someone talking in a different room, but their words are not clear, because of the distance or the walls. You might say "I can hear someone murmuring in the next room." Also, an animal or a machine can murmur, but they cannot mumble. I hope this helps.

There is a small difference in meaning between these two sentences. Both sentences suggest that it was difficult to understand his words. In that way the two sentences basically mean the same thing. However, "murmer" and "mumble" suggest different reasons for why it was difficult to understand his words. "Murmur" suggests that he spoke the words too quietly to be understood. "Mumble" suggests that he pronounced the words too badly to be understood.