Technically speaking, nothing. And even a diamond listed as an Asscher cut will say "square emerald cut" on its certification. This is because the original Asscher cut diamond was the world's first patented diamond cut, created by Joseph Asscher in the early 20th century. But the family's patent expired during WWII, making the cutting pattern up for grabs by the general public. While the style is still called the Asscher cut, they are no longer patented or regulated, and instead are really just square emerald cuts. Hence, one in the same.