White sugar is hygroscopic - it attracts water and locks it away from the dough to make your cookie crisp. Brown sugar, with it's added molasses content, keeps the dough softer and makes the cookies chewy. So if you like a crisp, crunchy cookie, use all white sugar, or adjust the proportion of brown sugar downwards.
In most baking recipes, you shouldn’t substitute brown sugar for white sugar. If your recipe calls for 1 cup white sugar, swap 1 cup brown sugar. The sweetness level will be exactly the same, but the brown sugar may change the texture of your baked goods. You’ll likely notice a more robust flavor and the color of the finished baked good may be darker as well. (Substitution is not recommended)
These alterations in color, flavor, and texture come from the way brown sugar is made. Brown sugar is white sugar with molasses mixed in; as much as 10% molasses, by weight, depending on the manufacturer. That molasses might make the texture moister, so it might be helpful to slightly decrease the amount of the wet ingredients (like milk or water) in your recipe or slightly increase the dry ingredients (such as flour, cocoa powder, or oats). You’ll also probably notice a hint of caramel or butterscotch flavor.
Brown sugar generally works much better in quick breads than light and airy cakes—we’re looking at you, angel food—since these take advantage of the lighter texture of the white sugar. Substituting brown sugar for white sugar will actually be a win if you prefer softer and chewier over crispier cookies since the molasses lends that extra moistness.
Dark brown sugar offers a stronger molasses flavor while light brown sugar has a milder flavor, but structurally, the two sugars will work the same. The difference in the amount of molasses is so minimal, no one will likely notice the difference.