We have had a solution from someone who unfortunately didn't give his or her name, but it says that there are $14$ different rectangles:

Rectangles 1 and 4 are the same as each other.
Rectangles 5, 8, 10 and 12 are the same as each other.
Rectangles 6 and 11 (which are squares) are the same as each other.
Rectangles 3 and 13 are also squares but they are not the same as each other. They are similar to 6 and 11.
Rectangles 2, 7, 9 and 14 are different to each other and not similar to any other rectangle.