To simplify the expression \( \sqrt{2+\sqrt{3}} - \sqrt{2-\sqrt{3}} \), we can use the identity for the difference of two squares:

Let \( a = \sqrt{2+\sqrt{3}} \) and \( b = \sqrt{2-\sqrt{3}} \).

Then the expression is \( a - b \).

The square of this expression is:

\[
(a - b)^2 = a^2 - 2ab + b^2
\]

Calculate \( a^2 \) and \( b^2 \):

\( a^2 = 2+\sqrt{3} \) and \( b^2 = 2-\sqrt{3} \).

So:
\[
a^2 + b^2 = (2+\sqrt{3}) + (2-\sqrt{3}) = 4
\]

Now calculate \( ab \):

\[
ab = \sqrt{(2+\sqrt{3})(2-\sqrt{3})} = \sqrt{4 - 3} = \sqrt{1} = 1
\]

Plug these into the squared expression:

\[
(a-b)^2 = 4 - 2(1) = 2
\]

Thus, \( a-b = \sqrt{2} \).

Therefore, the simplified form is \( \sqrt{2} \).