Two-dimensional graphene/molybdenum disulfide (Graphene/MoS₂) van der Waals (vdW) heterostructures, which integrate the strong optical absorption of MoS₂ with the ultrahigh carrier mobility of graphene, have emerged as promising candidates for low-power, high-sensitivity photodetection. Although previous studies have primarily focused on photoresponse metrics such as responsivity and specific detectivity, the fundamental characteristics of dark current and zero-offset voltage have received comparatively limited attention. In this work, a Graphene/MoS₂ vdW heterostructure photodetector was fabricated by mechanical exfoliation followed by an all-dry transfer process. The dark current and zero-offset characteristics were systematically investigated over a temperature range of 100-300 K under 532 nm laser illumination with power densities ranging from 28 to 226 mW cm-2, and were directly compared with those of a commercial Si PIN photodiode. The Graphene/MoS₂ heterostructure exhibited an ultralow dark current on the order of 10^-12-10^-11 A, approximately two orders of magnitude lower than that of the Si PIN photodiode under identical measurement conditions. This substantial reduction is attributed to the suppression of dark-state carrier transport by the interfacial potential barrier in the vdW heterostructure, whereas the dark current in the Si PIN device is predominantly governed by the thermal generation and collection of carriers within the depletion region. Zero-offset measurements further revealed a temperature-dependent polarity transition in the Graphene/MoS₂ device, with the offset voltage changing from negative to positive at low temperatures and rapidly converging toward 0 V as the temperature increased. This behavior indicates the dynamic modulation of the interfacial electrostatic potential by localized potential fluctuations and trap-state charging/discharging processes. In contrast, the Si PIN photodiode consistently exhibited a larger positive zero-offset voltage over the entire temperature range, with its variation primarily determined by shifts in the net current equilibrium within the p-n junction. These results provide new insights into the intrinsic dark-state transport and offset mechanisms of Graphene/MoS₂ vdW heterostructure photodetectors, offering valuable guidance for the design and optimization of low-noise, high-stability optoelectronic devices.