Let's solve the expression step by step:

The expression given is:

\[
\cfrac{1}{ 1 + \sqrt{5}} - \cfrac{1}{ 1 - \sqrt{5}}
\]

To simplify, we'll start by rationalizing the denominators of each fraction.

Step 1: Rationalize the denominators

For the first term, \(\cfrac{1}{ 1 + \sqrt{5}}\):

Multiply both the numerator and denominator by the conjugate of the denominator:

\[
\cfrac{1 \cdot (1 - \sqrt{5})}{(1 + \sqrt{5}) \cdot (1 - \sqrt{5})} = \cfrac{1 - \sqrt{5}}{1^2 - (\sqrt{5})^2} = \cfrac{1 - \sqrt{5}}{1 - 5} = \cfrac{1 - \sqrt{5}}{-4} = \cfrac{\sqrt{5} - 1}{4}
\]

For the second term, \(\cfrac{1}{ 1 - \sqrt{5}}\):

Similarly, multiply both the numerator and denominator by the conjugate of the denominator:

\[
\cfrac{1 \cdot (1 + \sqrt{5})}{(1 - \sqrt{5}) \cdot (1 + \sqrt{5})} = \cfrac{1 + \sqrt{5}}{1^2 - (\sqrt{5})^2} = \cfrac{1 + \sqrt{5}}{1 - 5} = \cfrac{1 + \sqrt{5}}{-4} = \cfrac{-(1 + \sqrt{5})}{4} = \cfrac{-1 - \sqrt{5}}{4}
\]

Step 2: Subtract the fractions

Now, subtract the two fractions:

\[
\cfrac{\sqrt{5} - 1}{4} - \cfrac{-1 - \sqrt{5}}{4}
\]

Since the denominators are the same, we can combine the numerators:

\[
\cfrac{(\sqrt{5} - 1) - (-1 - \sqrt{5})}{4} = \cfrac{\sqrt{5} - 1 + 1 + \sqrt{5}}{4} = \cfrac{2\sqrt{5}}{4} = \cfrac{\sqrt{5}}{2}
\]


The simplified expression is:

\[
\cfrac{\sqrt{5}}{2}
\]

Thus, the correct answer is:

\(\textbf{B. } \cfrac{1}{2} \sqrt{5}\)