Ultrafast demagnetization of Co/Pt heterostructures induced by a femtosecond 800-nm laser pulse launches a spin current from Co to Pt and subsequent conversion of the spin current to a charge current in the Pt layer due to the inverse spin-Hall effect. At the same time, due to the spin-dependent photogalvanic effect, a circularly polarized femtosecond laser pulse also generates a photocurrent at the Co/Pt interface. Both ultrashort photocurrent pulses are effectively detected in a contactless way by measuring the THz radiation they emit. Here we aim to understand how the properties of the Co/Pt interface affect the photocurrents in the bilayers. By varying the interfacial roughness, crystal structure, and interfacial intermixing, as well as having an explicit focus on the cases when THz emissions from these two photocurrents reveal opposite trends, we identify which interface properties play a crucial role for the photocurrents. In particular, we show that by reducing the roughness, the THz emission due to the spin-dependent photogalvanic effect reduces to zero while the strength of the THz emission from the photocurrent associated with the inverse spin-Hall effect increases by a factor of 2. On the other hand, while intermixing strongly enhances the THz emission from the inverse spin-Hall effect by a factor of 4.2, THz emission related to the spin-dependent photogalvanic effect reveals the opposite trend. These findings indicate that microstructural properties of the Co-Pt interface play a decisive role in the generation of photocurrents.