The special things about sugar cookies is the extra softness you get from each bite. The reason behind that are the ingredients which include; sugar and other sweeteners. All the categories of dry sugar (granulated, brown, powder or confectioners) are hygroscopic in nature, they have a strong attraction when mixed with water.
The basic advantage they provide is that they safeguard the structure builders (starch and proteins present in flour and egg) from moisture and water. So the end product results in sugar cookies having a shelf life longer than normal cookies; along with maintaining the same tenderness, softness and moistness as when you first baked them. Just think of the situation in which you reduce the amount of granulated sugar, the result will be a larger amount of water content in the sugar cookies. This in turn will make the cookies crunchy because the dough is stiffer and won't spread.
When considering the crunchiness or softness of the sugar cookie you might also consider that the finer the granules the better it will be to spread the cookies on the table. Also, bear in mind that the confectionery sugar has more of cornstarch which also reduces the ability for you to spread the cookie dough around. If the sugar is added in proper proportion then it also stabilizes the whipped egg whites. This is because of the following:
The procedure of spreading out and the collection of the egg proteins slows down.
It protects the air bubbles from collapsing by creating thick viscous syrup.
Also it is always good to know the sources of the granulated sugar so that we are in a better position of fusing the correct constituents. The source is generally the sugarcane or the sugar beets. Sucrose is the chemical name of granulated sugar. The common usage is in baking sugar cookies.
Confectionery sugar is mixture of cornstarch (3%) and granulated sugar. The finesse of the sugar ranges anywhere from 6x to 10x. Confectionery sugar cannot be used as a substitute of the granular sugar because of following reasons:
The air bubbles cannot be trapped
The sweetness is lesser
The cookies that are made from brown sugar will have a softer texture & a darker color. The difference arises during the baking process where the melting of sugar crystal takes place & it moves to the surface. It is there where the dark color is provided and it caramelizes for added flavor. Adding or blowing air can be done for creaming softened butter with sugar.