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.
alla breve, cut time: used for marches and fast orchestral music. Frequently occurs in musical theater. Sometimes called "in 2", but may be notated in 4.
That really isn't the case, that is the job of the tempo, a BPM speed set by the composer. The intended difference is purely that 3/4 groups quavers into groups of two, 6/8 into groups of three.
