Hey,
basically you've got different sized snare wire to allow you to get the ratio of Drum to Snare sound that you want.
For example; a snare with a 12 strand wire is going to have much more "drum" and "shell" sound, theoretically making it a more sound more open.
Where as a 42 strand wire will cover up a lot more of the resonator head and give you more "snare" sound, theoretically making it drier. And as ComposerofRequiems said it can make a snare more sensitive because there are more wires to resonate
Obviously this isn't always the case because of the different ways we tune, the length of the snare beds, and our own idea of what is a 'dry' sound or a 'wet' sound etc, etc...
Before you go changing up your snare wire width just check that snare beds on the drum are wide enough to accomodate the wire you're looking at using. You should have at least an inch of snare bed on each side of the wire to allow it to seat on the head properly
Dunno if i explained that very well
Hope it helps