Non Turing-Recognizable from symmetric difference
(Converting the comment hints and guidance into an answer)

If you take $A$ to consist of all strings, $A=\Sigma^*$, then $A$ is decidable and recognizable. 
Then, we will take the second language, $B$, to be recognizable but not decidable. 

Their symmetric difference will give us (by definition) $\overline B \triangleq \Sigma^* \setminus B$.

Now, let's get concrete. The halting problem, $HP$, is a recognizable non-decidable language. If we take $A=\Sigma^*$ and $B=HP$, then their symmetric difference is $\overline {HP}$. The last step is to show that $\overline {HP}$ is not-recognizable (but this is a very well known fact that $\overline{HP}\in coRE$ and not-recognizable, since if it was recognizable, then $HP$ would have become decidable, since for any input we would have a machine that says "yes" if it is the language, and another machine that says "no" if it is not).