In general, the longer or bigger the resonant cavity, the narrower the resonance. However, you also typically get multiple resonances. So, one advantage of a ring resonator is narrower linewidth. Otherwise, the advantage depends on the application I suppose.

thank you sir. What more can we expect from ring resonator structure in filters? Almost 99% of Papers on photonic crystals based filters are designed with ring resonators? any specific reason?

In photonic crystals the resonant structure needs to be much less than a wavelength across. This makes it very difficult to use cavities. Cavities were used in early metamaterials (70 years ago) such as in the magnetron resonator, backward wave oscillators and cavity travelling wave tubes, and can be used under microstrip, or in the walls of waveguides, for instance, but these are all basically one-dimensional metamaterials/photonic crystals.

Thank you Malcolm... While using ring resonators do we have specific calculation or mathematical equation to find its resonant frequency/ wavelength of the photonic crystal structure?

A ring resonator is resonant when the mean circumference is a wavelength. A line resonator is resonant when it is a half-wavelength long. If it is bent in a circle the end capacitance increases so a C resonator can be less than half a wavelength in circumference. There are lots of papers on the internet for free you can read about this - read 10 or 20 and check their maths and you should find out what you need.