\[\sqrt{24 - 6\sqrt{15}} - \sqrt{115 - 20\sqrt{15}} =\]
\[= \left| \sqrt{15} - 3 \right| - \left| 10 - \sqrt{15} \right| =\]
\[= \sqrt{15} - 3 - 10 + \sqrt{15} =\]
\[= 2\sqrt{15} - 13.\]