There are a few things that keep Saturn's rings roughly the way they are.
First, Saturn's D ring actually is "raining" down on Saturn currently. But, the phenomenon of shepherd moons prevents the vast majority of material from leaving the other rings: "The gravity of shepherd moons serves to maintain a sharply defined edge to the ring; material that drifts closer to the shepherd moon's orbit is either deflected back into the body of the ring, ejected from the system, or accreted onto the moon itself." (quote from Wikipedia)
Besides this, the majority of the particles within the ring system have almost no motion towards or away from Saturn; no motion towards the planet prevents them from being lost.
Second, Saturn's rings cannot clump into "full-fledged" moons, but they can clump into moonlets up to several hundred meters to a few kilometers across. At last count, I think there were over 200 that had been found, and they also come out of numerical simulations.
Beyond these larger moonlets, quasi-stable clumps and clusters of ring particles form with great frequency the farther you get from Saturn. These clusters of particles are constantly changing size, trading material, etc., and so there's no time for them to become solid and cohesive.
This gets into the idea of the Roche Limit and Hill Spheres. The basic idea of the Roche Limit is that the closer you are to a massive object, the more tidal forces are going to tear you apart (or prevent you from forming to begin with). Hill spheres are related, where the idea is at what point you're gravitationally bound to one object or another. If you're within Saturn's Hill sphere versus a moon's Hill sphere, you're going to be pulled to Saturn. With both concepts, you'll need to have a moon forming farther away from Saturn than its rings are now to actually be stable.
You can see the effects of these by looking at N-body dynamical simulations of the rings. This was my research for a year and a half, and it culminated in over a hundred simulations, many of which I made movies of, and then I posted them on one of my personal websites. If you go to it, scroll down and take a look at one of the C ring simulations, B ring simulations, and A ring simulations (warning - the movies are a bit big). You should choose ones with a large τ value and ρ of 0.85 because those will show clumping better.
What you'll see is that, in the C ring, almost no clumping occurs. Go farther from Saturn into the B ring and you'll see a spider web start to happen of strands of clumps of particles. Then if you go to the farther away A ring the strands are fragmented more into clusters. (Note on the movies: The "L" value next to each one is how large the simulation cell is on a side, in meters. So you're just looking at a VERY small region of the ring. It's set so that the center of the cell doesn't move, so you'd imagine that whole thing orbiting around Saturn.)
Best Answer
There seems to be a known explanation. I quote from Composition, Structure, Dynamics, and Evolution of Saturn’s Rings, Larry W. Esposito (Annu. Rev. Earth Planet. Sci. 2010.38:383-410):
I think the key is that particles in a thick ring would not move in parallel planes but would have slanted trajectories, colliding all the time and losing their energy very fast.