In this article, we study quasilinear Volterra integro-differential equations (VIDEs). Asymptotic estimates are made for the solution of VIDE. Finite difference scheme which is accomplished by the method of integral identities with using of interpolating quadrature rules with weight functions and remainder term in integral form are presented for the VIDE. Error estimates are carried out according to the discrete maximum norm. It is given an effective quasilinearization technique for solving nonlinear VIDE. The theoretical results are tested on numerical examples.