The biggest difference I find with "electric" live music is dynamics, bass impact, and how the room is energized. Clarity, detail and other common audiophile desires are pretty far down the list when trying to recreate this sound in my living room. As long as it's not too loud (which is a problem in smaller venues) I never hear anything harsh, fatiguing or bright.
With other kind of live music, its the dynamics and how great the tone is, especially in the midrange. I also notice decay hanging in the air a lot more than the leading edge having any great qualities.
I understand why people who are really into the gear side of this hobby like the ultra detailed sound. If you buy a $3,000 cartridge, you really want to hear the difference between it and the $1,000 one you just replaced. Which is cool, not trying to knock anyone and I'm sure people are having lots of fun doing that sort of gear worshiping. I find it a little amusing though that they never admit it and they always say its about the music. But to me, dynamics, natural tone and lovely decay is what Im trying to achieve.