The function $f(x)$ satisfies
\[b^2 f(a) = a^2 f(b)\]for all real numbers $a$ and $b.$  If $f(2) \neq 0,$ find
\[\frac{f(5) - f(1)}{f(2)}.\]
Setting $a = 5$ and $b = 2,$ we get
\[4f(5) = 25f(2),\]so $\frac{f(5)}{f(2)} = \frac{25}{4}.$

Setting $a = 1$ and $b = 2,$ we get
\[4f(1) = f(2),\]so $\frac{f(1)}{f(2)} = \frac{1}{4}.$  Hence,
\[\frac{f(5) - f(1)}{f(2)} = \frac{25}{4} - \frac{1}{4} = \boxed{6}.\]