Sugar cookies are made with butter, sugar, eggs, vanilla extract, flour, and leavening agents such as baking powder or baking soda. Shortbread cookies, on the other hand, are made with only a few simple ingredients: butter, sugar, and flour. They do not typically contain eggs or leavening agents. Sugar cookies are soft and tender, with a slightly chewy texture. Shortbread cookies, on the other hand, are crumbly and have a buttery, melt-in-your-mouth texture.