Steam boiler and a steam generator

A boiler, as the name suggests, boils the water before turning it into steam at subcritical pressure - the pressure at which bubbles can form. 

Steam generators, on the other hand, convert water into steam into steam without boiling at a super-critical pressure.

There are also constructional differences between a boiler and a steam generator.  A boiler contains many tubes which carry the water.  A steam generator has, generally,  only one tube in which the heating occurs.