Prescriptive tradition holds that correlatives should introduce parallel units, ie units of equivalent function.
Unlike Either ... or, both ... and, neither ... nor cannot link complete clauses.
Compare (the conjoins are in bold):
Neither he saw the film nor he saw the play. (conjoined clauses) :cross:
He neither saw the film nor saw the play. (conjoined predicates) :tick:
He saw neither the film nor the play. (conjoined Od's) :tick:
He neither saw the film nor the play. unparallel (predicate and Od conjoined) :cross: