\[\begin{array}{l}
\sqrt {{{\left( {2 - \sqrt 3 } \right)}^2}} - \sqrt {{{\left( {2 + \sqrt 3 } \right)}^2}} \\
= \left| {2 - \sqrt 3 } \right| - \left| {2 + \sqrt 3 } \right|\\
= 2 - \sqrt 3 - \left( {2 + \sqrt 3 } \right)\,\,\,\,\left( {do\,\,\,2 - \sqrt 3 > 0} \right)\\
= 2 - \sqrt 3 - 2 - \sqrt 3 \\
= - 2\sqrt 3 .
\end{array}\]