Several reference works have attempted to draw distinctions between need and necessity since Fleisher did so in 1804. From George Crabb, English Synonymes Explained, in Alphabetical Order (1816):
Necessity respects the thing wanted; need the person wanting. There would be no necessity for punishments, if there were not evil doers ; he is peculiarly fortunate who finds a friend in time of need. Necessity is more pressing than need : the former places in a positive state of compulsion to act ; it is said to have no law, it prescribes the law for itself ; the latter yields to circumstances, and leaves in a state of deprivation. We are frequently under the necessity of going without that of which we stand most in need.
From George Graham & Henry Reed, English Synonymes Classified and Explained (1847):
Need is exigent and pressing; necessity is stern and unyielding. Necessity demands; need requires. Those who are in necessity are in the lowest degree of poverty, and have no means of supplying their commonest wants; those who are in need are in a temporary difficulty, from which a moderate help will relieve them. Necessity forces us to act for ourselves; in our need, we require the assistance of our friends. We may manage to do without what is needful, but hat is necessary cannot be dispensed with.
From Charles Smith, Synonyms Discriminated: A Dictionary of Synonymous Words in the English Language (1910):
NEED (A.S. nead, want, compulsion) relates directly to the urgency of the demand, and indirectly to the absence of supply. Want is commonly absence of mere possession; need, absence of means of action. As they express states, NECESSITY (Lat. necessitatem) is stronger than NEED, for whereas NEED is negative, NECESSITY has a positive and compelling force. A man is in need of food. Under some circumstances there is necessity for immediate action. Need is pressing, necessity unyielding. Need is the strongest degree of requirement, necessity of demand. In the phrase of the English Psalms, "See that such as are in need and necessity have right," the second term is an augmentation of the first. Need may be temporarily and easily removed; necessity is more lasting and less remediable. We need, in cases of difficulty, the advice and support of friends; but lacking this, we are often impelled, by necessity, to decide and act for ourselves.
From James Fernald, English Synonyms, Antonyms, and Prepositions, thirty-first edition (1914):
Necessity is the quality of being necessary [elsewhere in the book defined as that "which must exist, occur, or be true; which in the nature of things can not be otherwise"], or the quality of that which can not but be, become, or be true, or be accepted as true. Need and want always imply a lack; necessity may be used in this sense, but in the higher philosophical sense necessity simply denotes the exclusion of any alternative either in thought or fact; righteousness is a necessity (not a need) of the divine nature. Need suggests the possibility of supplying the deficiency which want expresses; to speak of a person's want of decision merely points out a weakness in his character; to say that he has need of decision implies that he can exercise or attain it. As applied to a deficiency, necessity is more imperative than need; a weary person is in need of rest; when rest becomes a necessity, he has no choice but to stop work.
From Webster's Dictionary of Synonyms (1942):
Need, necessity, exigency come into comparison when they denote a state or condition requiring something as essential or indispensable, or the requirement itself. Need implies pressure and urgency arising either from external or internal causes or forces: it may merely suggest the call of an appetite or demand for emotional or intellectual satisfaction (as, he is in need of food; the child suffers because of a need for affection; he felt the need of an education), or it may imply circumstances, such as great poverty, a severe storm, a threat of war, actual hostilities, or the like, that show a lack of or create a demand for something indispensable to the well-being, protection, security, success, or the like, of those concerned (as, to provide food and lodging for those in need); the European war has taught Americans the need for a two-ocean navy; "So at the threat ye shall summon—so at the need you shall send Men"—Kipling). Necessity, though often interchanged with need, usually carries a far stronger suggestion of an imperative demand or of a compelling cause; as, telephone me in case of necessity (i.e., if in very great need); "as soon as war is declared, every nation or institution must subordinate all other considerations to the necessity of victory" (Inge). "When a legal distinction is determined ... between night and day, childhood and maturity ... a point has to be fixed or a line has to be drawn ... to mark where the change takes place. Looked at by itself without regard to the necessity behind it, the line or point seems arbitrary" (Justice Holmes). Sometimes, however, in order to enforce this implication, necessity is qualified by some such term as compelling; as compelling necessity drove him to seek financial aid. Necessity may also apply to a compelling principle or abstract force inherent in nature or in nature in in the constitution of a thing and inevitable in it operations or inescapable in its results; as, there is no logical necessity apparent to the conclusions you have reached; physical necessity lies behind our nee of food and drink; "One of the unhappy necessities of human existence is that we have to 'find things out for ourselves'" (T.S. Eliot).
The updated version of this dictionary, Merriam-Webster's Dictionary of Synonyms (1984) uses different examples to illustrate the distinctions it makes between need and necessity, but the wording of the distinctions themselves is virtually unchanged from the 1942 edition.
From Bergen Evans & Cornelia Evans, A Dictionary of Contemporary American Usage (1957):
need; necessity; want Need and necessity are nouns which designate a lack or a demand which must be filled. Need, a word of Old English origin, has connotations which give it a strong emotional appeal) (A friend in need is a friend in deed. I had most need of blessing;, and "Amen" / Stuck in my throat. O, reason, not the need! Our basest beggars / Are in the poorest thing superfluous). Necessity, a word of Latin origin, is more formal and impersonal; or objective. Though much stronger than need in expressing urgency or imperative demand, it is less effective in appealing to the emotions (Necessity is the mother of invention. The art of our necessities is strange, / That can make vile things precious). ...
From William Morris & Mary Morris, Harper Dictionary of Contemporary Usage, second edition (1985):
necessity/need Necessity can be followed by either "of" or "for," as in "There is no necessity for (or of) alarming him at this time." An infinitive is never used after necessity; rather, need is used with an infinitive, as in "There is no need to alarm him at this time." Since necessity and need are synonymous, there is little logic to this. It is simply a matter of idiom.
An Ngram chart of "no necessity for" (red line) versus "no necessity of" (blue line) versus "no necessity to" (green line) for the period 1700–2019 suggests that "no necessity to" has been in idiomatic English use for centuries:
But the key point in Morris & Morris's treatment of the entry for necessity/need is the assertion that "necessity and need are synonymous." Coming after the numerous efforts to distinguish the two terms from one another, this matter-of-fact statement that the two words mean the same thing is a bit startling—but I think it is true for most English speakers.
Some nuanced speakers and writers seem clearly aware of a greater intensity, longevity, or motivating force of necessity in comparison with need. Others are more fitful in their awareness of any distinction in sense or overtone between the two words—or simply use them interchangeably.
In trying to distinguish between close synonyms, you should bear in mind that most people who grow up speaking a language acquire their understanding of what the words mean more or less osmotically—especially in the case of common words that they are likely to have encountered (and used themselves) from a fairly early age. They don't start out by consulting a dictionary and trying to apply the meanings they find there to the situation they want to talk about. For this reason, it is difficult to estimate with any confidence the extent to which real-world usage ever abided by the definitions and distinctions that lexicographers and usage experts recorded and prescribed through the years.