When enjoying live recordings we get more than just a great capture of the music. We get a sense of the crowd, the stage, the air conditioning system, the floor bounce, the room acoustics.
Live recordings are the polar opposite of the sterility of studio recordings where great pains are expended to neutralize studio acoustics, quiet the room, make whisper-quiet the air conditioning system, and silence any hint of an audience.
Live recordings capture both the brilliant musical surprises as well as whatever mistakes happen (which is one reason many live recordings are actually a compilation of multiple night performances presented as one).
In a studio session do-overs, punch-ins, editing, and overdubs cover the mistakes and often over sterilize the performance.
There are clear differences between the spontaneity of live recordings and the sterility of the studio versions. Both have their charms, benefits, and weakness.
The one observation I will make is that I wish more studio recordings would be alright with some environmental noises. When we were recording the Audiophile's Guide SACD reference disc, many tracks were captured in PS Audio's warehouse where, without much effort, one can clearly hear the air conditioning system at work, creaks, groans, and sighs from the metal roof, and a generally more "live" sound.
I often miss all that recording studios spend small fortunes eliminating.