Dark soy sauce has a richer, more complex, and less salty flavor compared to light soy sauce. It's also a bit sweeter and has notes of molasses and caramel. It's also thicker and obviously darker in color.
Dark soy sauce is sometimes used in cooking to add color, but most importantly for me as a chef, it adds a depth of flavor that light soy sauce does not. Light soy sauce is great for both seasoning (i.e. salt) and umami (glutamates) as well as its own flavor profile.
Dark soy sauce is fermented and aged much longer than light soy sauce. This extended aging increases complex flavor development and gives it a thicker consistency and a darker color. Think of 25-year DOP balsalmic vinegar compared to the crap they try to pretend is from Modena in your grocery store.
Dark soy sauce isn't typically used as a table sauce like light soy sauce, but try it on scrambled eggs. Yum.