The timing of quartz clocks is regulated by a crystal oscillator. This crystal oscillator effectively forms an RLC circuit. If this is so, what properties does a crystal oscillator have that makes it advantageous over an RLC circuit?
A quartz crystal is a mechanical resonator with particularly stable properties. Quartz is a very stable material -- it doesn't 'age', or change much with temperature. It is also possible to prepare quartz to be very pure and have consistent properties. Quartz is also slightly piezoelectric -- an electric field causes a deflection, and a deflection generates an electric charge.
When cut correctly (with a specific orientation w.r.t the crystal axes) and mounted correctly, the mechanical properties (basically stiffness) are independent of temperature. Contacts on the crystal mean that a mechanical vibration generates electrical charge, and when configured correctly (with an amplifier), the whole system can be made resonate at a stable frequency.
Electrically this can be modeled as a RLC network with similar properties. The RLC values may be surprising -- typically fractions of a fF of capacitance and many henries of inductance.
The reason is accuracy. For capacitors 2% is considered a very good tolerance. I'm not sure about inductors but I expect it's similar. Resistors are better than capacitors or inductors but you can't build an oscilator with resistors alone.
To put these numbers in perspective: 1% is equivalent to 36 seconds per hour or 14 minutes and 24 seconds per day, which would be totally unacceptable accuracy for a clock.
While a quartz crystal can be modeled as an RLC circuit, that's not what it actually is.
The cut & dimensions of the crystal cause it to be resonant at a particular frequency and this can be much more accurately determined than a circuit made of discrete R's, L's & C's.
From my experience, a crystal is added instead of replacing the RLC components of an oscillator. The reason it is "added," is to give and maintain a given frequency more accurately than using the RLC components alone. The reason a crystal provides more accuracy, is that it can be manufactured to "tighter" tolerances than the RLC components and its high Q electrical property.