The in ear monitor has drastically changed the way bands can play music. At first the bands would just have big speakers at their feet playing what the band was playing (yes, live.)
Then with mix desks, the band members were able to have certain sounds at their feet louder than others. It may be important for Bono to hear Larry more than it is the Edge, to keep the beat of the song.
With the in ear monitors, they can do just about anything. I know that Larry often listens to "click-tracks" which are basically metronomes. This gives him a clicking sound to the beat of the song they are playing. Some say this is cheating, but some say it just makes the band sound better. I believe the latter. Imagine having to start the songs of a 21 year career with different beats and tempos all the time.
So yes, they are listening to live music(what they are playing), but they listen to different mixes in their ears. I read somewhere what each band member listens to, but forgot where.
I know Adam does not like to use them at all.
During the Elevation tour you would notiice that there were no speakers on the "heart." Due to this the band needed in ear monitors to hear themsleves when they went out there. Because Adam does not use them, he would not go out onto the heart, except for the acoustic sets where he would be standing next to Larry to hear him.
Now you can see why a mix at an awards show would be so hard. Imagine U2 playing after, let's say, Metallica. The mix guy would have to know what each band member in each band wanted to listen to inside thier ears.
This is also why you many times see Bono (or anyone else in a band) look at their mix guy and yell things like, "more guitars" or "less drums" and many times they actually point to their ears.
Lastly many musicians use these to simply listen to themselves play. If Bono was standing on stage singing and playing let's say an acoustic guitar, he may just want to listen to his guitar over the rest of the music.
I think the whole thing is very interesting.