The difference is the side on which pressure is regulated, and the reaction curve.
Typical pressure regulator takes arbitrary input pressure (from compressor) and provides specified (lower) pressure on output, locking the valve as output pressure rises; not allowing it to drop below preset level. The vacuum regulator assures the input pressure is as specified, while accepting arbitrary output (vacuum pump), by closing as the input pressure drops to preset. In the regulator feedback loop, the "readout" is performed on different sides of the valve. Since the regulators are usually unidirectional devices, not allowing for backflow, you can't just flip one around, and expect it to work like the other - if e.g. you put a vacuum pump in place of the compressor connected to a tank through a standard pressure regulator - it won't work.
You can imagine a tank that needs to maintain a pressure in specific range; the fill input will have a pressure regulator that allows filling the valve to preset pressure. But if for other reasons (e.g. the tank being heated by sunlight) the pressure rises, the relief valve will disengage, releasing the excess build-up; a vacuum regulator works similar to the relief valve, except its set point is somewhere below 1 bar, while a typical relief (safety) valve will open when internal pressure rises considerably above atmospheric.