Soaps and detergents are both surfactants; They decrease the surface tension of water--in effect making it wetter. 

Soaps

Soaps are made by mixing oils or fats with lye which was originally produced from wood ash.  These ingredients react to make the basis for soap.  Properly made and cured soap has no lye and the remaining oils provide a comfortable protective coating for the skin.  These oils result in a product that with a ph balance that is slightly acidic and matches the ph of human skin. 

Benefits: 

Soaps are much more skin friendly than detergents.  While the difference matters to all skin types, children, and those with sensitive skin, eczema, psoriasis, dry or aging skin will notice the most difference.  Using soap flakes in place of laundry detergent can help prevent babies' rashes.

Drawbacks:  The residue that soap leaves may benefit skin, but dulls the shine of hair when used as shampoo, and may be less effective than detergent at cleaning laundry.

Detergents

Detergents are synthetically manufactured.  They were first developed in the 1940's and have since replaced soap in almost all commercial cleaning products. 

Benefits:  Detergents rinse almost completely leaving no film to dull the shine of hair.  They make excellent shampoos and laundry detergents for those who do not have extremely sensitive skin.

Drawbacks: Detergents are slightly alkaline unless tempered with extra oils or acids and are disruptive and drying to human skin.   Detergents increase skin permeability speeding drying and allowing irritants easier entry.  They have been indicated as irritants in eczema and psoriasis flares.