What is the purpose of this adjustment? Why should I use it over just converting to grayscale?
The black & white adjustment has multiple options for tuning how you want to transform colors into purely grayscale. It seems most apt for converting color photos into black and white. If converting a photo, you will notice the powerful control of details in the scene and be able to apply contrast/levels right there on the fly. Simply hitting Mode - > Grayscale will not allow you to fine tune visual elements that may require your hand, or at least your eyes to be aware of, and instead will use the default Black & White settings.