Answer: $\Large\frac{20^{24}}{24^{20}}$ is the bigger expression.
To see this, we first look at the prime factorizations of the numerators and denominators in each expression:
Let $A\Large = \frac{15^{16}}{16^{15}} = \frac{3^{16}\cdot 5^{16}}{2^{60}}=\frac{3^{36}\cdot 5^{16}}{2^{60}\cdot 3^{20}}$ and $B\Large = \frac{20^{24}}{24^{20}}=\frac{2^{48}\cdot 5^{24}}{2^{60}\cdot 3^{20}}$
Now consider $\frac{20^{60}\cdot 3^{20}}{5^{16}}A = 3^{36}$ and $\frac{20^{60}\cdot 3^{20}}{5^{16}}B = 24^8\cdot 5^8$.
Observe that $3^{36}=3^{16}\cdot 3^{20}=9^8\cdot 3^{20}$.
On the other hand, $2^{48}\cdot 5^8=2^{40}\cdot 2^8\cdot 5^8=4^{20}\cdot 10^8.$
Clearly $4^{20}>3^{20}$ and $10^8>9^8$, therefore, $4^{20}\cdot 10^8>9^8\cdot 3^{20}$.
Combining the results above, we have shown that
$\Large\frac{20^{60}\cdot 3^{20}}{5^{16}}$ $B\Large>\frac{20^{60}\cdot 3^{20}}{5^{16}}$ $A$ or equivalently, $B>A$
as desired.