A mono sound system is a system where all of the audio signals are mixed onto and sent through one audio channel. From a listener's perspective, all the elements of the music will sound as if they come from one particular space as opposed to a more scattered effect. In a live setting, this can be advantageous as each member of the audience will be able to hear all the instruments in the mix.
A stereo sound system is a system which has two independent audio channels. Each of the audio signals will have a level and phase relationship to each other, mixed in a way that is intended to portray the placement of instruments on a stage. For example, when mixing two guitars, the sound of the guitarist standing to the left of the stage would be panned slightly in favor of the left audio channel. This is a beneficial setup as it allows the sound to replicate what the audience can see on stage.
Live mono mixing doesn't produce a sound as realistic as stereo, because the sound won't replicate the placement of the instruments on stage. Mono can also sound a little bit squashed, especially when there are a lot of instruments, vocals and effects in the mix.
Stereo mixes are a little more difficult to create. More often than not, an audio mixer will end up in a situation where there's a "sweet spot" between the two outputs. The term "sweet spot" refers to an area where everything can be heard; by comparison, anywhere outside this area, one speaker can be heard more than the other. In a live setting, this is a big disadvantage as the majority of the audience--those who are not in the "sweet spot"--won't hear everything. There are ways to solve this problem; strategically placing the mics and speakers, for example.
Both mono and stereo have their advantages. A mono mix is a lot easier to manage live as a mixer can guarantee the whole audience will be able to hear everything. With stereo mixes, on the other hand, it takes a lot of time and effort to create a perfect mix without losing quality in certain areas. In saying that, a good stereo mix is often considered better than a good mono one.