Live Music, to me, as an event, should be a full sensory experience: your ears AND eyes should be treated to a SLEW of stimuli, ranging from the music itself to a light and video show (if applicable), or at very least, watching the musicians on stage, not just playing the music, but PERFORMING it, with a live intensity not SEEN (but sometimes heard and felt) in studio recordings.
Now, as a strictly musical aspect, with regards to Studio Vs. Live, I think Live music offers a chance for the performer(s)/musician(s) to present their music in a different light, either with different instruments, instrumentation, arrangements, expanded sections, longer solos, improvised jams, etc. etc., giving their fans a (somewhat) unique experience. Plenty of bands do this while others seem to prefer to offer live music in a format that is closer to their studio counterparts, but in a live setting. RUSH have been known to try to replicate studio songs in a live setting, and much of the time, they do - their 2nd live album Exit...Stage Left is a very 'cold' and 'dry' album in terms of being live. The songs are presented with little-to-no crowd noise (cheering, roaring, clapping, singing along), and barring a few variations to the songs, they're almost like the studio versions. However, later live releases amp-up the live feel, such as their Rush In Rio release, which features a crowd of over 40,000 Brazilians singing along to songs, even instrumentals, and cheering louder than any other crowd Rush has performed in front of.
Now, some bands usually offer shows that are (almost) ALWAYS unique, with plenty of live jams and improvs. Jam bands probably have a huge following, with millions of bootlegs and bootleggers, mostly because each show IS different from each other. In that case, live music definitely has it's charms, and their fans either love or hate it (because it can be expensive/time-consuming to get EVERYTHING).
I think for many bands, there are some great live albums, such as Yes' Yessongs, where the songs are taken to the next level with bigger arrangements, longer jams, spontaneous solos, etc. etc., and where the music is transformed both by the band and the audience. There are some live albums, however, which don't offer much in the way of being different than their studio counterparts.
-Marc.