To find the subtle difference between “mumble” and “murmur”, let’s first review the official dictionary definitions of these two words.

Mumble

1.To utter indistinctly by lowering the voice or partially closing the mouth: mumbled an insincere apology.

2.To chew slowly or ineffectively without or as if without teeth.

verb, intransitive

1.To speak words indistinctly, as by lowering the voice or partially closing the mouth.

2.To chew food slowly or ineffectively, as if with the gums.

noun

A low, indistinct sound or utterance.

Excerpted from The American Heritage® Dictionary of the English Language, Third Edition  © 1996 by Houghton Mifflin Company.

Murmur

murmur (mûr´mer) noun

1.A low, indistinct, continuous sound: spoke in a murmur; the murmur of the waves.

2.An indistinct, whispered, or confidential complaint; a mutter.

3.Medicine. An abnormal sound, usually emanating from the heart, that sometimes indicates a diseased condition.

verb murmured, murmuring, murmurs verb, intransitive

1.To make a low, continuous, indistinct sound or succession of sounds.

2.To complain in low mumbling tones; grumble.

verb, transitive

To say in a low indistinct voice; utter indistinctly: murmured his approval.

Excerpted from The American Heritage® Dictionary of the English Language, Third Edition  © 1996 by Houghton Mifflin Company.

From these definitions, we could see that “to mumble” is to speak in an indistinct manner as if the mouth were half closed, or as if there were some chewing in the mouth. The words would come out in spurts.

To murmur is to speak like in a whisper, in a continuous manner, with a full mouth slightly open.