There are two main advantages to CRDS over other absorption methods:
First, it is not affected by fluctuations in the laser intensity.  In most absorption measurements, the light source must be assumed to remain steady between blank (no analyte), standard (known amount of analyte), and sample (unknown amount of analyte).  Any drift (change in the light source) between measurements will introduce errors.  In CRDS, the ringdown time does not depend on the intensity of the laser, so fluctuations of this type are not a problem. Independency from laser intensity makes CRDS needless to any calibration and comparison with standards.
Second, it is very sensitive due to its long pathlength.  In absorption measurements, the smallest amount that can be detected is proportional to the length that the light travels through a sample.  Since the light reflects many times between the mirrors, it ends up traveling long distances.  For example, a laser pulse making 500 round trips through a 1-meter cavity will effectively have traveled through 1 kilometer of sample.
Thus, the advantages include:
*High sensitivity due to the multipass nature (i.e. long pathlength) of the detection cell.
*Immunity to shot variations in laser intensity due to the measurement of a rate constant.
*Wide range of use for a given set of mirrors; typically, ±5% of the center wavelength. 
*High throughput, individual ring down events occur on the millisecond time scale.
*No need for a fluorophore, which makes it more attractive than laser-induced fluorescence (LIF) or resonance-enhanced multiphoton ionization (REMPI) for some (e.g. rapidly predissociating) systems.
*Commercial systems available.