The answer is simple, hold a ruler or any straight edge along the diagonal line in each diagram and you will see that one is slightly convex, and the other is slightly concave. Neither one has a perfectly straight line from point to point and the difference is enough to add up to the surface area of one square.

The angles of a perfect right triangle that is 13x5 would be about 68.96° and 21­­.04°. The angles of the red triangle (8x3) are actually 67.98° and 22.02°. The dark green triangle's angles (5x2) are 68.2° and 21.8°.

The area of the red triangle is 12, green is 5, which is 17. The area of the whole "triangle" should be 32.5 but because of the difference in angles it's actually only 32. When you switch their positions, the shapes that form the square in the base are 8x2 or 16 so you have an extra square (16+17=33 in the bottom one). This would not happen if the lines were drawn perfectly straight and ignoring the graph paper's boundaries.

It is possible to create a perfect triangle using the squares on graph paper, such as a 4x4, but not with 13x5. I don't know if the fact that there are lots of prime numbers in this diagram have anything to do with it or not.

Another way to look at it is copy the top triangle and rotate it so it forms a rectangle, you would see an opening where they meet. That opening should add up to about one half of a square. The other one would overlap in the same amount, or the difference between one square and the space left in the top one. Either way they add up to one square.