I think it's steam pressure if you are talking about a device, machine, ...However, vapour pressure can also be used in other contexts, for example in talking about the changing phase of liquid to vapour. Hope it helps.
Use steam pressure when talking about steam (gaseous water). Vapour should be able to be used in either case, but definitely when the pressure comes from a substance other than water.
Distillation is a physical operation of separating the mixture of several components/or multicomponents mixture on the basis of the different steam/vapour pressure and boling-points of the individual components into: base petrol (50-220°C), petroleum for illumination (180-280°C), grease oils and paraffins (more than 350°C) and nonevaporating remnant(bitumen and coke).
Distillation is a physical operation of separating a multicomponent mixture on the basis of the different vapour pressures and boling-points of the individual components into: base petrol (50-220°C), fuel oil/diesel oil (180-280°C), grease oils and paraffins (more than 350°C) and involatile residues(bitumen and coke).