Brown Sugar is table sugar with molasses. The molasses adds the brown color to the sugar and is a byproduct of partially refining or not refining sugar from sugar cane. Brown sugar is also created by adding molasses directly to refined white sugar. This gives the producers of brown sugar more control over the amount of molasses present in the product.
Brown sugar, generally, has a much smaller particle size that granulated white sugar. Particle size effects the final product of baked goods that utilize the creaming method. Particle sizes controls the size of the air pockets in a baked good.
Another consideration that effects food is the acidity. Molasses is moderately to extremely acidic meaning that it has the ability to greatly impact the final product. It has the ability to denature proteins and react with bases to create CO2. These considerations can be positive or negative depending on the acidity that the batter or dough already has before introduction and personal preference.