What is in a recording? What is the data present in it?
The musicians and the venue.
So, I think if we ignore the musicians for a bit, this question really can be re-written this way:
How much of the recording environment comes through the sound system?
That is, the entire difference between the two questions posed by the OP is really the same as this question, above. To be clear, unless we were present, this really is going to be an unknown quantity. Lots of ways to give the illusion of being in a performance, but that doesn't make the illusion accurate. So I think the answer to this feeling, not necessarily truth, is in the speaker dispersion and room acoustics.
Is the room transparent enough to let the recording of the venue reach our ears with clarity? Does the speaker work with or against the room? Does the speaker have rear ambience drivers?
Now, which do we like... that's another story. :)