Because the micro-pulse of 6/8 is different than 3/4. 3/4 is naturally emphasized as "1 2 3 1 2 3. In contrast, 6/8 is naturally emphasized with a micro pulse of 1 2 3 4 5 6, which produces a strong pulse on 1, a slightly weaker pulse on 4, and a macro pulse of "1 2. (Think 4/4 with triplets). You thus get a much different feeling from 6/8 than 3/4.

Primarily for reasons of legibility, in my opinion.

Technically, any piece in duple meter could be written in any duple meter and any triple meter piece could be written in triple meter. In fact, you could even write a triple meter piece in duple meter if you are willing to notate everything as a triplet.

Also, as some of the answers below show, part of legibility is in supporting the speed or the feel of the piece.

(Also 2/2 is a fairly common time signature better known as cut-time. Many marches, for example, use this.)

Mostly to indicate tempo and style. A 3/4 measure and a 6/8 measure are the same length, but the 6/8 has twice as many beats at twice the rate. So a composer would use a 6/8 signature for a faster-moving piece with a lot of eighth notes. Similarly, a slow piece would have a 2/2 signature instead of 4/4.

6/8 really has two beats in a bar, each beat containing three quavers, while 3/4 has three beats in a bar.