We numerically study soliton dynamics at the interface between media with uniform and periodically modulated self-focusing nonlinearities. We find that the soliton can spontaneously laterally drift if its power is large enough. The drift direction can be controlled by changing the sign of the nonlinear modulation coefficient. We also study the dynamics of soliton launched with a tilt angle toward the nonlinear interface and reveal unique features, such as soliton rebound, penetration, and trapping.