The reasons are plentiful.
It's dangerous (cancer, respiratory/cardiac arrest, or death, anyone?)
It's time-consuming (indefinite storage? we're talking several millennia at best, and by then it'll be our future generation's problem to deal with).
Reprocessing makes it easier for Plutonium to make its way around the world.
The effects of radioactive waste can ruin meadows, water supplies, and the ground from which this water comes from.
As of yet, we have no permanent solution to the nuclear waste problem. Until it can be found, I suggest we cease our reliance on the nuclear power plants of America.