Example 2 Calculate the value of $\sqrt{14+6 \sqrt{5}}-\sqrt{14-6 \sqrt{5}}$ is ( ).
(A) 1
(B) $\sqrt{5}$
(C) $2 \sqrt{5}$
(D) 5
(2000, National Junior High School Mathematics League)
Let $\sqrt{14+6 \sqrt{5}}=a+b$,
$$
\sqrt{14-6 \sqrt{5}}=a-b .
$$
(1) $\times$ (2), (1) $)^{2}+(2)^{2}$ respectively yield
$$
\begin{array}{l}
a^{2}-b^{2}=4 \text {, } \\
a^{2}+b^{2}=14 \text {. } \\
\end{array}
$$
(4) - (3) gives $b^{2}=5$.

Since $b>0$, thus, $b=\sqrt{5}$.
Therefore, the desired expression $=(a+b)-(a-b)=2 \sqrt{5}$.