In this paper the authors apply the general procedure of the Khovanov-Lauda-Rouquier type quantum affine Schur-Weyl duality in order to obtain an exact tensor functor from the category \(\mathcal A\) of finite-dimensional graded modules over the symmetric quiver Hecke algebra of type \(A_\infty\) to the category \(\mathcal C_{B_n^{(1)}}\) of finite-dimensional integrable modules over the quantum affine algebra of type \(B_n^{(1)}\). This functor is shown to factor through a certain localization \(\mathcal T_{2n}\) of the category \(\mathcal A\), which is related to the categories of finite-dimensional integrable modules over the quantum affine algebra of type \(A^{(t)}_{2n-1} \), \(t = 1, 2\). This gives rise to a ring isomorphism between the Grothendieck rings of appropriate categories of modules over quantum affine algebras of types \(A\) and \(B\), which induces a bijection between the sets of classes of simple objects.