work vs engagement

work and engagement both are nouns.

Word	Noun	Adjective	Verb	Adverb
work	Yes	No	Yes	No
engagement	Yes	No	No	No

As nouns, engagement is a hyponym of work; that is, engagement is a word with a more specific, narrower meaning than work:

work: the occupation for which you are paid
engagement: employment for performers or performing groups that lasts for a limited period of time

work (noun)	engagement (noun)
activity directed toward making or doing something	the act of sharing in the activities of a group
the occupation for which you are paid	a hostile meeting of opposing military forces in the course of a war
a product produced or accomplished through the effort or activity or agency of a person or thing	the act of giving someone a job
a place where work is done	employment for performers or performing groups that lasts for a limited period of time
the total output of a writer or artist (or a substantial part of it)	a mutual promise to marry
applying the mind to learning and understanding a subject (especially by reading)	contact by fitting together
(physics) a manifestation of energy; the transfer of energy from one physical system to another expressed as the product of a force and the distance through which it moves a body in the direction of that force	a meeting arranged in advance

work (verb)	engagement (verb)
give a workout to
arrive at a certain condition through repeated motion
go sour or spoil
cause to undergo fermentation
find the solution to (a problem or question) or understand the meaning of
use or manipulate to one's advantage
to mix into a homogeneous mass
perform as expected when applied
move into or onto
shape, form, or improve a material
cause to happen or to occur as a consequence
prepare for crops
make something, usually for a specific function
gratify and charm, usually in order to influence
proceed along a path
move in an agitated manner
provoke or excite
be employed
cause to work
exert oneself by doing mental or physical work for a purpose or out of necessity; work
have an effect or outcome; often the one desired or expected
operate in a certain place, area, or specialty
proceed towards a goal or along a path or through an activity
have and exert influence or effect
operate in or through
cause to operate or function
behave in a certain way when handled