What is the different between these two triangles? What is the different between rigorous proof and proof based on intuition on this problem?
It seems to me that these triangle are equivalent in area.

 A: A picture is worth $\aleph_\omega$ words. Here are the two triangles one over the other.
See how they don't quite have the same outline?

$\hspace{120pt}$

A: Notice that, in both triangles, you have a change in the slope of "hypotenuse".
The second one, completed with the missing square, would not be a triangle anyway, as the first is not!
Or, better, notice that: $\tan^{-1}\left(\frac{2}{5}\right)\ne\tan^{-1}\left(\frac{3}{8}\right)$. The catheti of the blue and red triangles are respectively $2$ and $5$, $3$ and $8$ and their hypotenuses are not parallel. Yeah, that's all!
A: Easiest place to see it, for me, is to start from either the bottom left or top right of each triangle, and count five squares horizontally and two vertically (the size of the blue triangle). You'll see that in each case one of the figures hits the grid point and the other one doesn't.
Yes, you could predict this as Matt says by noting that the red and blue triangles aren't similar, so the figures are not triangles at all. I already knew that having seen the trick before, so I went for pure visual "spot the difference" instead. The grid itself helps to establish the illusion by clearly showing the sizes of the smaller pieces, but it also helps dispel it.
