Trust your identities

Let $a=\dfrac {16}{21}$ and $b = \dfrac 37$ . Then $ab = \dfrac {16}{21} \times \dfrac 37 = \dfrac {16}{49}$ . And

$\begin{aligned} \frac {\left(\frac {16}{21} \right)^3-\left(\frac 37 \right)^3}{\left(\frac {16}{21} \right)^2 - \blue{\frac {16}{49}}} & = \frac {a^3-b^3}{(a+b)^2 - \blue{ab}} \\ & = \frac {(a-b)(a^2 + ab + b^2)}{a^2 + 2ab +b^2 - ab} \\ & = \frac {(a-b)(a^2+ab+b^2)}{a^2+ab+b^2} \\ & = a - b = \frac {16}{21} - \frac 37 = \boxed{\frac 13} \end{aligned}$