The main difference is that when you use a CW laser, if the sample has any absorption, you will heat the sample, and a change of temperature produces a change in refractive index. Therefore, you will be measuring a thermally induced nonlinearity, which are usually large (can be observed with only a few mW powers) and respond fairly slow (micro to milliseconds). Pulse lasers (ps-fs) on the other hand, allow observing fast responding electronic nonlinearities, but care has to be taken to avoid the presence of thermal contributions, particularly with the relatively long ns pulses
You can use both CW laser as well as pulsed laser for the measurement of third order susceptibility. In CW laser thermal effects will be a significant factor