After watching the Food Network for 10 yrs, and having a Daughter as a professional chef, here is the difference....
White sugar will add 1 thing and 1 thing only to a recipe, Flavor... Brown sugar and other less refined sugar will impart the individual flavor of any residual ingredients they carry. I usually use dark brown sugar in rubs as it is moister and allows the rub to pack a bit better.
If you like to experiment, find different sugars like Demerara (yes like the rum) and see what works for you. I only use refined white sugar in a recipe that I need the product to remain uncolored, or I don't want to deal with the packing effect of a packed sugar.
Just my 2 cents worth.
EDinNB