Actually though, with a stage play, there typically
is a conductor: for the stage crew. Everybody on the crew is wearing a headset, and the person "calling show" is following a marked-up script to call spotlight cues (which spot to follow which actor, when to open the aperture, how BIG to open up), music cues, light board cues (even with a digital board, the timing of cues is manual), etc.
The difference to me is to do with synchronicity. Multiple lighting cues need to sync up with the action on stage, so you need someone to call show. A musical ensemble obviously needs to sync up rhythmically. Actors, however, don't typically need that kind of third-party direction during performance. If they do, generally the director will come up with some kind of offstage cue (a gunshot sound effect, for example) or designate one actor to subtly cue the other actors (touch her nose, for example).
The other thing is obviously just the nature of the performance. Enjoying a play, unlike music, requires a suspension of disbelief--an immersion in the setting of the show, a loss of awareness of your actual surroundings--and having the director standing in front, pointing to people and mouthing orders at them makes that impossible. Not to mention, having somebody up on stage waving their arms around is just plain distracting for a visual experience, but not at all for an auditory one.