'Cause it’s really frigging expensive to send a probe to Saturn? Cassini is the first and only probe that we’ve put in orbit around Saturn. It is one of the largest and most advanced probes ever built, and it launched on the heaviest rocket available at the time. Getting it to Saturn required no less than four gravitational assist maneuvers over seven years. It’s cost is over $3 billion so far.
AFAIK, NASA doesn’t have the budget or the political will to even think about launching a comparable mission any time soon.
However, you might get you want at the end of the Cassini mission. Once the probe is nearly out of fuel they’re planning to crash it into Saturn, which seems like it would include a closer pass to the rings.
As already pointed out, we have sent the Cassini probe there, and it has provided a lot of observations of the rings, including the particle size distribution.
I guess you’re asking why we haven’t taken really close-up photos or chemical analysis. First of all, that would require the spacecraft to go into the same orbit as the ring particles, which is much lower than Cassini’s. It would take a lot of fuel to go into such an orbit, likely a lot more than Cassini has. (A probe is coming in from outside the Saturn system, so getting to a low orbit requires a lot of energy.) And getting close enough to photograph individual particles safely would probably require autonomous navigation. For chemical analysis, you need to get even closer.
But I think the more important issue is, if we’re going to send a probe all the way to the Saturn system, what’s the most useful thing to do with it? Study the rings, which every evidence says is just rock and dust? Or study Titan, which has a thick atmosphere and even liquid oceans (of methane)? Or Enceladus, which has liquid water under the ice? Or Saturn itself, which has complex storm systems, magnetic fields, etc? Looking at the proposed Titan Saturn System Mission, it seems clear that the rings are a low priority among scientists.
Also, scientists who do study the rings are mostly interested in the large-scale behavior. It’s a fantastic natural laboratory for orbital dynamics and collisional effects. But you don’t want close-up photos for such studies, you want high-resolution remote sensing, not in-situ measurements at a few spots.