I can see as a listener in a crowd how one may not be able to tell many of these details. Now if it were a show with two or more groups, I'd bet one would be able to tell differences between gear of the different players. Maybe not specifically what, but that there are differences. I can usually tell if a player is playing flats or rounds, with a pick, often if through a GK or Ampeg, etc.
I've never been on the bandwagon of which fretboard wood sounds best though. Unless you're a fretless player, the string is vibrating on a fretted string between two pieces of metal, or an open string between metal and plastic, or bone, or whatever your nut is made of. The rest of the signal though, from pickups to cords, to amplification, to how you strike the string, will make a difference in sound.
I'd say that it's very easy though to notice a difference in your own sound if something changes. Others may not, but we will.
Being from a fairly small local market, I have often been walking into a club, and been able to name the player that happens to be playing inside, without knowing the name of the group on the bill. Some of that has to do with the sound, and of course some has to do with the style.