It's not a bad question, if we were just playing snare drum. Different areas of drumming use different sticking methods. Concert snare drummers, for example, basically always alternate and rarely use rudimental stickings. They certainly practice them, but they don't use them in performance.
On the drum set, stickings directly affect what you can play on the instrument, and how.
The right hand plays accents, the left hand plays very quietly, and each hand plays its notes on a single conveniently placed drum.
Now try doing the same thing with the same dynamics using an alternating sticking.
Apart from how the sticking looks on paper, that's a much more complicated affair. It's an extremely difficult way to play that idea, and in fact if you only used alternating sticking, you would never play that. You could conduct a similar experiment using the patterns from the beginning of Stick Control. See how many things that are easy to do with a simple mixed sticking, that are just about impossible to play alternating.
There's also a natural drum set orchestration where the right hand plays the cymbal (often with the bass drum in unison) and the left hand plays the snare drum. Obviously that would be pretty dull with just a RLRL sticking. Try the paradiddle sticking again, with your right hand on the hihat, or the ride cymbal, with some added bass drum:
And again try to do the same thing with an alternating sticking:
It's rather awkward doing this using the hihat, and it's quite impossible to do it with the ride cymbal.
As a rule: Stickings with combinations of singles and doubles are easier to play fast, and there are vastly more possibilities for moving around the drums. They'll naturally have a little texture to them; they have built-in accents, and the doubles will tend to sound different from the singles. It takes developed technique and deliberate execution to make them sound as even as straight singles.
Alternating singles are good where an even sound and solid rhythm is desired, where you want to reinforce the rhythmic grid, like in pop, rock and funk. When playing faster, singles are better when you want perfect evenness... though that takes developed technique, too. Many under-developed players habitually accent with their right hand, a la Wipeout. Drags and rolls are naturally associated with alternating singles.