The process in which a relatively unspecialized cell acquires specialized features of a collenchyma cell. This is a plant cell in which the primary cell walls are unevenly thickened, with most thickening occurring at the cell corners. Cells are living and able to grow, they are elongated, and lignin and secondary walls absent. Collenchyma cells make up collenchyma tissue which acts as a supporting tissue in growing shoots, leaves and petioles. This tissue is often arranged in cortical ribs, as seen prominently in celery and rhubarb petioles.