We use the fundamental swap property of logarithms: $a^{\log_b c} = c^{\log_b a}$.

Let's apply this identity to the first term, $3^{\log_4 5}$: By swapping the base of the exponent ($3$) and the argument of the logarithm ($5$), we get: $3^{\log_4 5} = 5^{\log_4 3}$.

Thus, the expression $3^{\log_4 5} - 5^{\log_4 3}$ simplifies to: $5^{\log_4 3} - 5^{\log_4 3} = 0$.