The short answer is: Asscher cut diamonds are square (ish), Emerald cut diamonds are rectangular (ish).
The reason I say “ish” is that both Assher and Emerald have the corners truncated which gives them eight sides around the girdle.
But a Square Emerald is exactly the same as an Asscher cut diamond.
Cuts like Asscher and Emerald have their outlines arranged parallel to the girdle. This method is called a step cut though occasionally it may also be referred to as a trap cut. These types of cuts emphasize the diamond’s clarity, whiteness, and polish. Therefore it is advisable to select diamonds with the highest grades of color, clarity, and polish.
Other step cuts include carré, baguette, lasque, trilliant/triangle, kite, lozenge, trapeze/trapezoid and obus.
When performing a diamond search you will usually find Asscher diamonds by checking the “Square” shape. The actual square shape had already been taken and named “Princess”, so shapes such as the Asscher shape were given the title “Square”, purportedly to simplify things. I agree that it doesn’t really simplify it that much, but there you are. At least with the internet it is possible to unravel all the mysteries of diamond terminology. It just takes a little research and patience. But you will also find Asscher cut diamonds in the Emerald category. If you look for Emerald cut diamonds which have a length to width ratio of 1:1 (or close to that like 1:1.05) then you’ll find Asscher cut diamonds.
History of the Asscher cut:
The Asscher cut was developed in Amsterdam in 1902 by one of the 20th century’s most prodigious diamond experts Joseph Asscher. The Asscher shape was the first signature cut to ever be patented and was under patent until shortly after WWII.
Trivia about the Asscher diamond cutters:
In 1907 and Joseph Asscher and Abraham Asscher were commissioned to cut the renouned Cullinan diamond which weighed over half a kilo (621 grams). Joseph Asscher contemplated for 6 months deciding on the best way to make the cut . When he was finally ready to begin cutting the diamond the first blow broke the blade. He took up the task again the following week and allegedly fainted after stiking the Cullinan with a tremendous blow. The Cullinan was cut into the Cullinan I and Cullinan II diamonds for the British royal family.