Vertically stacked van der Waals (vdW) heterojunctions of two-dimensional (2D) transition metal dichalcogenides (TMDs) have attracted a great deal of attention due to their fascinating properties. In this work, we report two important gate-tunable phenomena in new artificial vdW p-n heterojunctions created by vertically stacking p-type multilayer ReSe2 and n-type multilayer WS2: (1) well-defined strong gate-tunable diode-like current rectification across the p-n interface is observed, and the tunability of the electronic processes is attributed to the tunneling-assisted interlayer recombination induced by majority carriers across the vdW interface; (2) the distinct ambipolar behavior under gate voltage modulation both at forward and reverse bias voltages is found in the vdW ReSe2/WS2 heterojunction transistors and a corresponding transport model is proposed for the tunable polarity behaviors. The findings may provide some new opportunities for building nanoscale electronic and optoelectronic devices.