We can say something on such matrices $B$ by characterizing its eigenvalues, which coincide with the eigenvalues of $A$.
Since $A$ is a real symmetric matrix, it has real eigenvalues. Hence a necessary condition on $B$ is that it has real eigenvalues. But one can say more. There are necessary and sufficient conditions known for real numbers $\lambda_1,\ldots ,\lambda_r$ to be the eigenvalues of a non-negative symmetric matrix. See for example the paper of M. Fiedler, Eigenvalues of nonnegative symmetric matrices here: http://www.sciencedirect.com/science/article/pii/0024379574900317.

Update: We still have the additional condition that the diagonal of $A$ is $(a,a,\ldots ,a)$, but this is also discussed in the paper of Fiedler. Given $2n$ real numbers $\lambda_i$, one can say whether there is a nonnegative symmetric matrix with eigenvalues $\lambda_1,\ldots ,\lambda_n$ and diagonal $\lambda_{n+1},\ldots ,\lambda_{2n}$.