There are two extremes in trying to notate juggling (or dance, or any other ill-defined artistic endeavor); exhaustive detail and bare minimum. Siteswap is of the latter type. At its most austere, it remembers only one aspect of the juggling experience: the order in which balls are thrown and caught.
This has the following benefits:
1. Every juggling pattern has an associated siteswap, since in every pattern the balls are thrown and caught in some order.
2. Two patterns with different siteswaps are almost always very different from a juggling viewpoint. (This will not be true for exhaustively detailed labeling systems, which will sometimes remember detail better forgotten.)
3. On the other hand, while two patterns with the same siteswap may be considered very different by most jugglers, the differences can generally be stated as information additional to the siteswap.
Mainly, though, siteswap is particularly nice from a mathematical point of view; for example, it is very simple to check in one's head whether a list of throws is a valid pattern, how many balls are involved, what they do, whether the pattern is left-right symmetric, whether one can dive right into the pattern or if a sequence of transitional throws is needed... while this information is implicitly to be found in any representation of the throwing & catching order, it is most easily dug out of the siteswap.
Another sign of the naturality of siteswap is that it's very easy to get a handle on how difficult a pattern is; look at the highest number each hand has to do.
For an example, take the pattern 5 1, the 3-shower. The large number 5 suggests that this pattern will not be easy - in particular, substantially more difficult than the 3-cascade. (Which it is, contrary to what most non-jugglers guess; this in itself should be taken as a sign of the naturality of the notation.) The more hopeful idea is that if you can do this pattern, you'll be able to do 5. This certainly isn't true - in particular, if you can only do this pattern right-handed, then it only says that your right hand can do 5. So to learn 5 you should generally learn to asynchronously shower in 3 both directions.
A brief history of siteswap (and references to published articles)