Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this talk, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula (BDF2) with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels and a cut-off numerical technique, the unique solvability and corresponding maximum-norm error estimates of the high-order nonlinear difference scheme are established. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Optimal fourth-order in space and second-order in time mum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.