Why does Saturn have a large ring system, but Earth does not?

There is evidence that recently, within the last few million years, a moon wandered into Saturn's Roche limit and broke apart, creating the ring system. Namely, some of the small inner moons have unstable orbits, destined to eventually impact Saturn, and cannot be very old. Also, the icy moons have high albedos, indicating they have not yet collected much dust on their surfaces, and suggesting they are relatively young. It is likely that Saturn's ring system is only temporary, and will become rarefied over the next few million years.

One reason why Saturn's ring system is so pronounced is because it has geologically active moons that spew new material into orbit around Saturn, which helps replenish it. The Earth, on the other hand, doesn't have a geologically active moon nor does Earth have the mass required to break apart moons that might wander in too close.

Tidal forces (related to gravity and distance) destroy objects orbiting too close. The moon is far beyond that limit.

Rings, generally, are fleeting... they don't last very long. Saturn has the advantage of having many different moons - 60-plus, in Saturn's case - and there are 'shepard moons' in the plane of the rings that keep them 'together'. If those moons weren't there - Saturn's ring system would likely resemble those of Jupiter or Neptune.