First, let's talk about pressure...
Pressure is generally the result of molecules within a gas or liquid, impacting on their surroundings, usually the walls of the containing vessel. Its magnitude depends on the force of the impacts over a defined area. It is measured in ‘newtons per square metre’, given the special name ‘pascal’. There is also the traditional (but obsolete) unit ‘pounds force per square inch’ or 'psi'.
The relationship between pressure (p), force (F) and area (A) is given by: p=F/A
This equation applies whether the pressure is very small, such as in outer space, or very large, as in hydraulic systems.
So the word pressure is correct when referring to the entire range of ‘force per unit area’ measurements, although at extremely low pressures the concept of molecules exerting a force becomes more abstract.
So, what is vacuum?
The definition of a vacuum is not precise but is commonly taken to mean pressures below, and often considerably below, atmospheric pressure.
It does not have separate units and we do not say that ‘vacuum equals force per unit area’. Thus, strictly speaking, we do not need to talk about both pressure and/ or vacuum because vacuum is pressure. But the differences are often misunderstood and thus leaving out the word vacuum can falsely imply that the pressure in question is above that of atmospheric pressure.
Another definition of the distinction between pressure and vacuum comes from the industries which use and make pressure and vacuum equipment.
Broadly, if the force on the walls of the containing vessel is sufficient to permit its measurement directly, we are dealing with pressure technology but if the force is too small for direct measurement and has to be indirectly inferred, we are in the realm of vacuum technology.
This definition is not entirely self-consistent though; for example there is a class of instrument which operates in the vacuum region by measuring the deflection of a diaphragm.
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