I am not positive why I am doing this subject first but it is the one most fresh in my mind. Most professional players only have one kit, but they will have a multitude of snares. Why is this you ask? Well a professional kit will generally be able to cope with most genres, and with the help of studio engineering can make it perfect too. For those first stepping into the studio a basic kit and a snare is usually fine. Changing the tension of the snare and snare wires can greatly change the sound. However there comes a time when you want to make professional quality recordings and have lovely expensive microphones. At this point you want a sound that can only be appreciated when recorded through expensive microphones as they can pick up all the complex tones of such a drum. Usually most people would be very protective of such drums and as such keep them for the studio and instead keep a separate snare for live purposes. By this point you will have 2 snares, one really complex for the studio and one that is sturdy for constant live use. In the studio a ringy snare is best. You can dry it out with moongel, gaffa and towels but you cant add those complex overtones. For this usually you will want a metal snare, the best for this would be a seamless brass snare, but a standard brass snare is just as good. Some people value definition and crisp sound over ringy and such go for wooden snares. There is no issue with either, but it is down to personal preference of the player. However with such different sounds its good to have both, so at this point we are talking about owning 3 snares. Now for the average player this is enough, but for many other players or session drummers there are many other sounds to acquire. Take me for example, I do session work and it is not limited to Jazz. I have done rock, metal, folk, pop and hip/hop and as such have different snares for different scenarios.
12" Traps snare for hip/hop and pop. This snare is tight and crisp an gives off the standard dry crisp attack sound used in most popular songs today. A good example is Good Feeling-Flo Rida. This snare is good for both live and studio applications.
10" Wurlitzer snare. This snare is very old and due to its size very specialist. This snare very rarely gets any kind of use outside of an effects perspective and even then mostly on folk style songs.
13" Premier hammered brass. This is one of my go to snares for live. A bit too ringy for my personal taste in a studio and so mostly used live in hard rock and metal scenarios. A good example of its type of sound is Limp Bizkit-Take a look around.
14" Pearl Masters Birch. This is my go to studio snare. Often used with calfskin heads too. Very sensitive and great at any volume. For rock, some pop, blues and metal this is my go to studio snare. This standard wooden snare sound is heard on Quantum Factor-Planet X
14" PDP 20 ply Bubinga/maple snare. This is my other go to live snare. This is used in most of my gigs for its very dry sound and sensitivity. Used during anything that isn't heavy rock or metal. A good example of this type of snare is Nightwish-Last ride of the day
14" Boosey & Hawkes spun brass snare. This is an interesting snare. It is my favorite studio snare for jazz and live for jazz and brushes. It has very complex tones and plenty of ring. A good example of this sound of snare is Sing sing sing-Benny Goodman.
This way I can cover a multitude of sounds, however a lot of people add marching snares, 10" popcorn snares and so forth as ways of augmenting a kit live or in studio. A famous use of this is mike mangini. Hope this helps answer as to why you would want more than one snare. Honest answer? Its always good to be prepared, and always good to have an arsenal of different sounds to best compliment different sounds and genres.
EDIT: Recently the brass hand hammered has been traded in for an even better maple 14x4 (yep, the buddy rich size) and is now a part of the family. Pictures will come soon, until then enjoy this photo of my other snares :D