Stopping potential is well defined and equal to the photon energy minus working function of the photo-electric material. This potential level is sure to prevent any photo electron from approaching the anode, even the best fitted electron on a "freeway" to the anode and with velocity directed towards the anode.
Saturation potenial is more vague. This potential must ensure that all photo electrons will reach the anode in spite of possibly being directed to the glass tube, possibly collide with gas atoms, deflected by a cloud of other slow moving electrons and similar mishaps. This voltage depends on tube structure, gas pressure and similar characteristics.
Conversely, saturation current is again well defined hence usually being used with stopping potential.