Symbols: If but not only if

When people assert implications, they often implicitly involve universal quantification. For example, "if $n$ is a prime number greater than $2$, then $n$ is odd" really means "for all integers $n$, if $\dots$." When one denies an implication, one includes the universal quantifier in the denial, so it becomes an existential quantifier. For example, if someone says that "$n$ is odd" doesn't imply "$n$ is a prime number greater than $2$", he normally means to deny that "for all $n$, if $n$ is odd then $n$ is a prime number greater than $2$"; equivalently, he means to assert that "there exists an odd $n$ that is not a prime number greater than $2$." (Recall from propositional logic that the negation of $B\implies A$ is equivalent to $B\land\neg A$.) So your proposed combined connective, for implication in one direction and denial of implication in the other direction, will implicitly quantify the variables partly with universal quantifiers and partly with existential ones. This looks to me like a recipe for confusion and therefore well worth avoiding.

If, by good fortune, your statements $A$ and $B$ don't involve variables, so these quantification issues don't arise, then there is a fairly easy answer to your question. As I said above, the negation of $B\implies A$ is equivalent to $B\land\neg A$. Furthermore, this formula already implies that $A\implies B$, so $$ (A\implies B)\land\neg(B\implies A) $$ is equivalent to $B\land\neg A$. But remember, this use of propositional logic is legitimate only if your $A$ and $B$ don't involve any variables that are implicitly quantified in your implications.

It is doubtful that a symbol exists; I do not believe it is common usage.

Note that your situation is equivalent to "$A$ implies $B$, but $B$ does not imply $A$". There are many, many situations in mathematics when this is the case. For instance:

Independent random variables have zero correleation coefficient, but a zero correlation coefficient does not imply that the random variables are independent.

In fact, the distinction that $A \implies B$ does not imply that $B \implies A$ is so important that it is almost always best addressed with more than a basic symbolic representation. The reader demands to know why the converse does not hold! Examples of situations where the converse does not hold are almost always useful.