I graduated as a computational linguist. The cognitive sciences really weren't alluded to much at all at my uni. so can't help with the compare and contrast.
So, what a computational linguist does: Not being exhaustive here... computational linguists work with corpora, work to make corpora, design tools for corpora, like POS-taggers, word sense disambiguation; and design natural language parsers, work towards machine translation and text categorization eg. "is this spam or not", and I'd say the first word net sorts under CL.
I have a feeling computational linguistics (CL) might be to linguistics what AI is to computer science, developing methods, which when they are commonly in use, are no longer considered to be part of CL/AI. Which is okay, since CL/AI is already working on the next big thing.
So, basically whatever is bleeding edge language data crunching at the moment.
I'm assuming the reason your advisor wants you to do some CL is because you'll need to know the tools, corpora and methods. Then there's getting a better understanding of what's easy and not easy for a computer vs. what's easy and not easy for humans, which should be relevant to cog. ling. indeed, and finally, it's an obvious suggestion since CL is basically going all CS on poor, unsuspecting language data: it's fun! :)