The condition for  is $\overrightarrow{A}+\overrightarrow{B}=\overrightarrow{A}-\overrightarrow{B}$ is 

Let  $\overrightarrow{A},\overrightarrow{B}$ are represented by adjacent sides of parallelogram such that  $OP=\overrightarrow{A},OR=\overrightarrow{B}$

Then $\overrightarrow{OQ}=\overrightarrow{A}+\overrightarrow{B},\overrightarrow{PR}=\overrightarrow{A}-\overrightarrow{B}$

We get both diagonals equal  $\overrightarrow{A}+\overrightarrow{B}=\overrightarrow{A}-\overrightarrow{B}$

(i) if  $\overrightarrow{A}=\overrightarrow{B}$ and (ii) Angle between $\overrightarrow{A}$ and $\overrightarrow{B}$ is 90° 

Trivial solution is $\overrightarrow{A}+\overrightarrow{B}=\overrightarrow{A}-\overrightarrow{B}$ then $\overrightarrow{B}=0$