Solution:

By checking all the possibilities, one can show that $T$ has height $4$ and base lengths $4$ and $5$. Orient $T$ so that the shorter base is on the top.

Then, the length of the cut parallel to the bases is $\frac{4+5}{2}=\frac{9}{2}$. Thus, the top two pieces are trapezoids with height $2$ and base lengths $2$ and $\frac{9}{4}$, while the bottom two pieces are trapezoids with height $2$ and base lengths $\frac{9}{4}$ and $\frac{5}{2}$. Thus, using the area formula for a trapezoid, the difference between the largest and smallest areas is

$$d=\frac{\left(\frac{5}{2}+\frac{9}{4}-\frac{9}{4}-2\right) \cdot 2}{2}=\frac{1}{2}$$