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.
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.
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.
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.
The crystal oscillator have properties of cheapness in term of circuit complexity and unit price that makes it advantageous over an RLC circuit. RLC circuit requires more parts and adjustments. When properly designed and calibrated, a RLC clock is as accurate as a crystal oscillator clock. It's all about the cost and size.