The process in which a relatively unspecialized cell acquires specialized features of a sclerenchyma cell. A sclerenchyma cell is a plant cell with thick lignified walls, normally dead at maturity and specialized for structural strength. Includes fiber cells, that are greatly elongated; and sclereids, that are more isodiametric. Intermediate types exist. Cells may or may not be devoid of protoplasm at maturity. Cell form and size are variable.