In the way that these sorts of goals are typically defined, the difference between them is that stretch goals are much more ambitious than SMART goals.
One of the major attributes of SMART goals is that they are supposed to be attainable or achievable. That is what the "A" stands for. By contrast, stretch goals are supposed to be very difficult to achieve. They are supposed to make us have to "stretch" in order to have any chance of achieving them. Therefore, there is a much greater chance of failing to achieve a stretch goal than of having the same problem with a SMART goal.