I've read before that a good part of the reason that we don't hear
ourselves as others do is literally because we don't hear ourselves as
others do - others perceive our voice directly from mouth to ear. When
you speak, you hear yourself primarily through your skull and only
secondarily through your ear. It gives a completely false impression. I
play the sax and that has the same effect and maybe even stronger due to
the hard grip of the teeth on the mouthpiece. I always prefer to monitor
through headphones so I can hear the sound as others will hear it.