Here,

Side of the Larger square = 1 inch

Therefore Area Large square = $1^2 =1$

But we know Area of a square =$\frac{diagonal^2}{2}$

So 1 = $\frac{diagonal^2}{2}$

or diagonal = $\sqrt{2}$

And we know
smaller square region is $\frac{2}{3}$ the area of the larger square region

So it can be written as $\frac{(Diagonal smaller square)^2}{2}= \frac{2}{3} * 1 = \frac{2}{3}$ (Since the area of the Larger square = 1)

or $(diagonal of smaller square)^2 = \frac{4}{3}$

or diagonal of smaller square = $\frac{\sqrt4}{\sqrt3} = \frac{2}{(\sqrt3)} = \frac{2}{\sqrt3} * \frac{\sqrt3}{\sqrt3} = \frac{2\sqrt3}{3}$

Now,
Diagonal of the larger square is longer than the diagonal of the smaller square = $\sqrt2 - \frac{2\sqrt3}{3}$