I'm reading an article on liquid crystal lasers and they're saying periodicity determines lasing wavelength and cavity geometry determines emission wavelength. What is the difference?
There is not much difference, probably the context of where you found these terms will tell you the intent of the author.
In any case, emission of light is a property of materials, normally the emission spectrum is broad. Sometimes one might ignore the idea if a spectrum and the talk about an emission wavelength (singular) as a short hand to refer to the peak wavelength.
In lasers the lasing wavelength is the emission wavelength out of the whole emission spectrum which has the most gain. For this reason energy tends to funnel into the laser mode and it is virtually monochromatic.
Regarding the liquid crystals. I'm not an expert on that but they seem to be saying that the liquid crystal has a natural emission spectrum which can be tuned by changing the periodicity. The cavity (basically a wavelength selective mirror) will be designed to overlap with the broad emission spectrum and essentially picks the wavelength which has the most optical gain, therefore this becomes the lasing wavelength.