Summary: Let \(H^2(\mathbb{D}^2)\) be the Hardy space over the bidisk \(\mathbb{D}^2\), and let \(M_{\psi, \varphi} = [(\psi (z) - \varphi (w))^2]\) be the submodule generated by \((\psi (z) - \varphi (w))^2\), where \(\psi (z) and \varphi (w)\) are nonconstant inner functions. The related quotient module is denoted by \(N_{\psi,\varphi} = H^2(\mathbb{D}^2) \ominus M_{\psi, \varphi}\). In this paper, we give a complete characterization for the essential normality of \(N_{\psi, \varphi}\). In particular, if \(\psi (z)= z\), we simply write \(M_{\psi, \varphi}\) and \(N_{\psi, \varphi}\) as \(M_\varphi\) and \(N_\varphi\) respectively. This paper also studies compactness of evaluation operators \(L(0) | n_\varphi\) and \(R(0) | n_\varphi\), essential spectrum of compression operator \(S_z\) on \(N_\varphi\), essential normality of compression operators \(S_z\) and \(S_w\) on \(N_\varphi\).