A Basque cheesecake and a New York Style cheesecake differ in terms of origin, texture, and baking technique. A Basque cheesecake is originally from the Basque region of Spain and is known for its burnt and caramelized exterior, creamy interior, and rustic appearance. It is typically baked at a higher temperature for a shorter time, resulting in a more custard-like consistency. On the other hand, a New York Style cheesecake is dense, creamy, and smooth with a rich flavor. It is baked at a lower temperature for a longer time, giving it a firm texture.