What is the difference between making a cheesecake with confectioner's sugar vs. normal sugar?
While you could use it, it would likely affect the finished product. As a substitute in baking, when the recipe calls for granulated sugar, powdered sugar is not always suitable. And, although it is granulated sugar which has been processed into powder, cornstarch has also been added to prevent clumping. 

With sugar, the size of a granule determines how well air incorporates into the batter. The larger the granule, the more air and, since the batter needs that air to help it rise, using powdered sugar will not produce the same results.