The projected separation is an estimate of the minimum distance between two astronomical objects (separation) that are presumed to be are near each other, i.e., roughly the same distance from Earth, and are close to each other on the celestial sphere (i.e., not a large angle between them). It is simply a calculation of the distance between them if they were exactly equally distant from Earth, given the angle and their presumed distance. The separation estimate is no better than the estimate of distance to the objects.
Even with a good estimate of the distance to the objects, one of the objects may be closer than the other, so the actual separation could be an order-of-magnitude or more different. If the distance to the objects is known, it is a minimum possible distance between them.
Objects for which projected separations are of interest include visual binaries, galaxies in the same galaxy group or galaxy cluster, extra-solar planets from their host stars, and other objects presumed to be (possibly) near each other, such as if they appear gravitationally bound.