Is the difference of a non-recursive and recursive set recursive?
The empty set is recursive, hence B cannot be the empty set.

You are right the B-A is recursively enumerable.

You cannot prove B-A is not recursive for a good reason: it may be
recursive. Maybe you can imagine an example of A recursive, B
not recursive, and B-A recursive. Hint: it is not even necessary that
B be recursively enumerable, or A recursive.

Proof that $B-A$ can have properties more specific than recursively
enumerability (i.e. compatible with being recursively enumerable),
indpendently of the properties of B.


   Consider 2 disjoint alphabets $\Sigma$ and $\Sigma'$, such that
 $\Sigma\cap\Sigma'=\emptyset$. Take $C=\Sigma'\,^*$ and
 $A=\Sigma^*$. Let $E$ be a non-recursive language in $\Sigma^*$. Let
 $B=C\cup E$. Then $B$ is a non-recursive language on the alphabet $\Sigma\cup\Sigma'$, because $C$ is recursive (trivially). Now you can see that $B-A=C$ since substracting $A$ removes precisely $E$, i.e. all words in B that are in $\Sigma^*$. So you have $B$ non-recursive, and both $A$ and B-A are recursive. Here I chose $A$ recursive. But I could have chosen any non-recursive language on $\Sigma^*$ that contains $E$.


Similarly, you cannot prove anything about A-B. Think why.