Example 1 Find the range of $y=\sqrt{x^{2}-12 x+45}-$ $\sqrt{x^{2}-2 x+5}$.
Solution: The original expression is transformed into
$$
y=\sqrt{(x-6)^{2}+3^{2}}-\sqrt{(x-1)^{2}+2^{2}},
$$

where the fixed points are $A(6,3), B(1,2)$, and point $A$ is higher than point $B$.
$$
\begin{aligned}
\because & |A B|=: \sqrt{(6-1)^{2}+(3-1)^{2}}=\sqrt{26}, \\
& \left|x_{2}-x_{1}\right|=5,
\end{aligned}
$$
$\therefore$ The range is $(-5, \sqrt{26}]$.