Clarification is a name for the method of separating fluid from solid particles. Often clarification is used along with flocculation to make the solid particles sink faster to the bottom of the clarification pool while fluid is obtained from the surface which is free of solid particles.
Thickening is the same as clarification except reverse. Solids that sink to the bottom are obtained and fluid is rejected from the surface.
The difference of these methods could be demonstrated with the methods used in waste water processing: in the clarification phase, sludge sinks to the bottom of the pool and clear water flows over the clear water grooves and continues its journey. The obtained sludge is then pumped into the thickeners, where sludge thickens farther and is then obtained to be pumped into digestion to be prepared into fertilizer.