The band can get sick playing the song the same way for the hundred time and hence create a different arrangement of the song that reinvents the song that is superior to the album version
The band may play the song that is not marred by crappy production. Aka live performance are less likely to be affected by overcompression/loudness wars. Also live performances of 80's songs often sound better because they lacked the crappy 80's production that marred the original recording. The band may stripped down the overproduction from the album and perform work that is more raw and within he spirit of the song.
The band may be inspired by the audience and perform the song with more energy, passion that they couldn't replicate in a sterile studio environment.
The band may be inspired from an improvisational point of view and hence the notes they play in the guitar solo may be superior to what is present in the album
However, with all those elements, the live nature is only the means to the end result which is a superior performance/timbre/arrangement. If that same performance/timbre/arrangement was replicated exactly in the studio version then that studio version would be just as good as the live version. The fact that the band were unable to replicate it in the studio in practice doesn't change that it is theoretically possible to replicate it in studio.
So ultimately a what makes a good live performance is indistinguishable from what makes a good studio recording. Some bands may require a live environment to produce their best work and some band require a studio environment to produce their best work due to the unique individual personality of the musician but the end result would be just as good if it was perform live or in studio
Ultimately I treat live performance/album as indistinguishable to bands re-recording their song. Sometimes the re-recording is superior sometimes it isn't but the same elements that make a studio or live performance great are the same.
The only exception to the rule is U2 rattle and hum live version of Sunday Bloodt Sunday where the live nature is interwoven to the narrative of the song that can't be replicated in a studio but that is an exception rather the rule (I think U2 have a case of being one of the greatest live band of all time due to that)
So with the exception of U2 example there is no real reason to treat a live performance as something separate to studio release