Filtration operates entirely on particle or droplet size (and, to some extent, shape), such that particles below a certain size will pass through the barrier, while larger particles are retained on or in the barrier for later removal. The separating size is a characteristic of the barrier, the filter medium.
Sedimentation, on the other hand, operates on the density of the particle or droplet. Or, more correctly, on the density difference between the suspended particle and the suspending fluid. It is the force of gravity working on this density difference (or the much higher centrifugal force operating in a centrifuge) that causes separation by sedimentation – either of a solid from its suspension, or of a lighter solid from a heavier one.
Solid separating technologies have two prime purposes: the removal of unwanted solids from suspension in a fluid (which may itself be a wanted product or a waste that needs cleaning prior to discharge), and the recovery of a wanted solid product from its suspension (often following a prior crystallization or precipitation step). Either kind of equipment, filter or sedimenter, may be used for either of these purposes, although it is true that most solid recovery is achieved in filters or sedimenting centrifuges.