We obtain a simpler proof of a result in [J. Funct. Anal. 278 (2020), 108401]. By using an elimination method, we completely characterize the bounded and compact differences of two generalized weighted composition operators with the same order, i.e. \(C_{u,\varphi }^{(n)}-C_{v,\psi }^{(n)}\) , on the standard weighted Bergman spaces. Furthermore, we show a rigidity of the difference of two generalized weighted composition operators with different orders, i.e. \(C_{u,\varphi }^{(n)}-C_{v,\psi }^{(m)}(n\ne m)\) . The boundedness, compactness and Hilbert-Schmidt nature of \(C_{u,\varphi }^{(n)}-C_{v,\psi }^{(m)}\) are equivalent to those of \(C_{u,\varphi }^{(n)}-C_{v,\psi }^{(m)}\) . And the compact difference of the sum, i.e. \((C_{u_1,\varphi }^{(n)}+C_{u_2,\varphi }^{(m)})-(C_{v_1,\psi }^{(n)}+C_{v_2,\psi }^{(m)})\) is also studied.
