If you substitute brown sugar for the white sugar in the barbecue sauce recipe, it won't cause the sauce to be sweeter. It will give the sauce a bit more of a molasses flavor, since nowadays all brown sugar is, is refined white sugar that's been flavored with molasses syrup.
Here's a recipe I use that uses white sugar and molasses for sweetening and flavor. It would be fine to use brown sugar instead of white in this recipe.
KC BBQ Sauce
1-1/2 cup water 1/2 cup ketchup 1/3 cup white vinegar 1/4 cup tomato paste 1/4 cup molasses 2 tablespoons white sugar 1 teaspoon Liquid Smoke flavoring (hickory) 1/2 teaspoon onion powder 1/2 teaspoon black pepper 1/2 teaspoon Worcestershire sauce 1/8 teaspoon garlic powder 1/8 teaspoon cayenne pepper (optional)
