Atomic clocks keep time better than any other clock. They even keep time better than the rotation of the Earth and the movement of the stars. Without atomic clocks, GPS navigation would be impossible, the Internet would not synchronize, and the position of the planets would not be known with enough accuracy for space probes and landers to be launched and monitored.
Atomic clocks are not radioactive. They do not rely on atomic decay. Rather, they have an oscillating mass and a spring, just like ordinary clocks.
The big difference between a standard clock in your home and an atomic clock is that the oscillation in an atomic clock is between the nucleus of an atom and the surrounding electrons. This oscillation is not exactly a parallel to the balance wheel and hairspring of a clockwork watch, but the fact is that both use oscillations to keep track of passing time. The oscillation frequencies within the atom are determined by the mass of the nucleus and the gravity and electrostatic "spring" between the positive charge on the nucleus and the electron cloud surrounding it.