First, what are the differences? Brown sugar is really 95% white sugar with a thin layer of added molasses. It's usually removed from processing before it's completely refined, but not early enough to be called unprocessed by any means. And yes, brown sugar does contain trace amounts of minerals from the molasses but not enough to really benefit the body in any measurable way. The amounts are far too low.
What about Turbinado sugar (or raw sugar)? Turbinado sugar is made from the initial pressing of sugar cane. Since it's pulled out before the processing is complete, more of the natural molasses remains in the crystals. If we're strictly talking about calories, brown sugar and Turbinado sugar do rank a little lower than white cane sugar, but this is largely because they carry more moisture not because there's something inherently healthier about them.
So what to do? Can you use brown sugar or Turbinado sugar in your baking recipes instead of white sugar? My advice on this one is to be careful. Since both will contribute more moisture to your recipe and a different color and flavor profile, you want to only make this decision with a forgiving recipe like simple cookies or muffins. I wouldn't do this kind of substitution with delicate cakes, breads, or mousses.
How about just decreasing the amount of sugar in your recipes to make them less sweet in general? Again: be careful. The sweetness that sugar lends to a recipe is only one of its functions. Sugar also encourages the incorporation of air in a batter, aids in coloration, increases shelf life, and softens texture. So feel free to experiment and make adaptations, but start slowly. Do you ever bake with Turbinado sugar or use other alternative sugars? We'd love to hear about your experiences!
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(Images: The Truth About and via Flickr; Hain Pure Foods)