The combination of the two sugars together gives you the perfect combination of sugar to get the chewy texture you want for these. White sugar typically makes cookies crispier while brown sugar makes them chewy and dense; the two combined makes for the perfect cookie. If you don't have one of the sugars or if you just want chewy cookies, you could try just using one, it just won't give you the best results (you may end up with cookies that are a little crisp).