We get this call quite a bit at Fat Daddio's. The quick, simple answer is yes. But there will be a difference in your baked good's texture. Let me explain...
Baking is Science: Brown Sugar
In baking, sugar is more than a sweetener, it is a necessary element. Sugar adds volume, bulk and structure to baked goods. When you mix sugar with fat it adds air to the batter. Then when you add your leavening agent those air bubbles enlarge making your baking rise.
Brown sugar is a refined sugar just like white sugar. The brown color comes from the added molasses. I'm sure you've seen that brown sugar comes in different shades, which all reflect the taste of the molasses. Darker brown sugar has a stronger taste. The sweetness of brown and white sugar is the same. So substituting brown for white sugar will still give you the same sweet flavor. However, the darker the brown sugar the more molasses (or butterscotch) flavor you will taste.
QUICK TIP: Be sure to pack the brown sugar in your measuring cup, as white and brown sugar have different densities.
The texture of your baked good is what is affected most by the sugar substitution. When you replace brown for white you will get a denser, moister baked good. This is great for sweet breads such as banana or zucchini, but not so great for a cake. The brown sugar will tend to make your cake too wet. And brown sugar alone will not give you a thin, crisp cookie.
QUICK TIP: Store brown sugar in a glass or plastic jar in a cool dry place. If it hardens, add an apple slice for a few days and it will soften up.