True “soaps” are regulated by the Consumer Product Safety Commission and are not required to list their ingredients on the label. They are made up solely of fats and alkali. Many bar cleansers are not soaps, but contain synthetic detergent cleansing agents and various thickening agents that keep the bar in its bar form.
Most soaps are considered very drying due to their alkaline base (i.e., a pH over 8). Bar cleansers can be gentler than bar soaps, but are more often than not still drying, depending on their composition.
References for this information:
Cutis, December 2001, pages 12–19
Archives of Dermatologic Research, June 2001, pages 308–318
Dermatologic Clinics, October 2000, pages 561–575