Mumbling typically has a more negative connotation than murmuring. Mumbling means you are not enunciating your words clearly, and therefore it is difficult to understand you. Whereas murmuring may mean that you are in a quiet place, or in an intimate setting, where it’s appropriate to speak very softly. The two are not interchangeable.

I believe that using murmur in place of mumble, or vice versa, will result in confusion. That said, when in doubt, “murmur” is more universally applicable than “mumble.” Example: I murmur but I do not mumble.