First simplify the original expression:

$\left(\frac{4^9}{4^{15}}\right)^{1/3} = (4^{9-15})^{1/3} = (4^{-6})^{1/3} = 4^{-2} = \frac{1}{16}$

Now check each answer choice to find which is not equivalent to $\dfrac{1}{16}$.

Choice E: $(4^{24})^{1/3} = 4^{24/3} = 4^8 = 65,536$

Since $65,536 \neq \dfrac{1}{16}$, choice E is not equivalent to the original expression.