In obsolete|lang=en terms the difference between flourish and cherish
is that
flourish is (obsolete) to boast; to vaunt; to brag while
cherish is (obsolete) to cheer, gladden.
As verbs the difference between flourish and cherish
is that
flourish is to thrive or grow well while
cherish is to treat with tenderness and affection; to nurture with care; to protect and aid.
As a noun flourish
is a dramatic gesture such as the waving of a flag.