Webster's Dictionary of Synonyms (1942) includes both posit and postulate (as verbs) in a group of words that also includes presuppose, presume, assume, and premise:
Presuppose, presume, assume, postulate, premise, posit agree in meaning to take something for granted as true or existent, especially as a basis for action or reasoning.Their corresponding nouns presupposition, presumption, assumption, postulate, premise (or premiss), position when they denote that which is taken for granted or is accepted as true or existent are distinguishable in general by the same implications and connotations as the verbs. ... Postulate, either as a verb or as a noun, differs from assume or assumption only in being far more restricted in its application and more exact in its implications. One can assume or make an assumption at any point in a course of reasoning, but one postulates something or lays down a proposition as a postulate only as the groundwork for a single argument, or for a chain of reasoning, or for a system of thought. Postulate, therefore, has reference to one of the underlying assumptions, which are accepted as true but acknowledged as indemonstrable and without which thought or action (also artistic representation) is impossible because of the limitations of human knowledge or of human reason (or of art); thus, the ordinary man always postulates the reality of time and of space; the dramatist postulates certain conventions which it is necessary for the audience to accept; "belief in the uniformity of nature, which is aid to be a postulate of science" (B[ertrand] Russell); "the prevailing theological system is one which postulates the reality of guidance by a personal God" (A[ldous] Huxley); "the kind of curvature postulated by the generalised theory of relativity" ([James] Jeans). ... Posit and position differ from postulate chiefly in implying affirmation as a truth; they rule out the implication of an assumption which is often found found in postulate and substitute that of a declaration of faith or conviction. "Hooker...rests his positions on one solid basis, the eternal obligation of natural law" ([Henry] Hallam). "It seems to me an error to say that we have no grounds for positing the existence of God outside His relation to ourselves" ([William] Inge).
The gist of Webster's argument seems to be that both postulate and posit explicitly acknowledge something as assumed at the outset of an argument or an inquiry or an exposition of a system of action or conduct—but postulate makes no commitment to the fundamental correctness of the underlying assumption, whereas posit affirms it correctness more or less devotedly.
It is interesting to see how Merriam-Webster's Dictionary of Synonyms (1984) differs from its predecessor in a similarly long discussion of this group of words, starting with the fact that it identifies the noun form of posit as posit rather than position:
presuppose, presume, assume, postulate, premise, posit are comparable when they mean to take something for granted as true or existent especially as a basis for action or reasoning.Their corresponding nouns presupposition, presumption, assumption, postulate, premise, posit when they denote something that is taken for granted or is accepted as true or existent are distinguishable in general by the same implications and connotations as the verbs. ... Postulate, either as verb or noun, differs from assume or assumption in being more restricted in its application and more exact in its implications. One can assume or make an assumption at any point in a course of reasoning, but one postulates something or lays down a proposition as a postulate only as the groundwork for a single argument or for a chain of reasoning or for a system of thought. Postulate, therefore, has reference to one of the underlying assumptions, which are accepted as true but acknowledged as indemonstrable and without which thought or action or artistic representation is impossible because of the limitations of human knowledge or of human reason or of art {the ordinary man always postulates the reality of time and of space} {the dramatist postulates certain conventions which it is necessary for the audience to accept} {belief in the uniformity of nature, which is aid to be a postulate of science—[Bertrand] Russell} {the prevailing theological system is one which postulates the reality of guidance by a personal God—[Aldous] Huxley} {what I'm postulating in all this ... is that the unconscious, you see, has an enormous teleological sense—{Norman] Mailer} ... Posit, as a noun and verb, comes close to postulate in implying the laying down of a proposition as a base for an argument, a line of reasoning, or a system of thought, but it may differ in suggesting subjective and arbitrary grounds rather than, as postulate regularly does, objective and rational grounds for selection of the proposition {if she needs salvation, she will posit a savior—[George] Santayana} {he did not posit a world of wormless apples to set off the fruit he reported in such wonderful detail—[Peter?] Gratton} {materialism at that time posited the premise that character was the product of environment, and this was the basis of Zola's naturalism—[James?] Farrell} but even when it connotes actual falsity it remains very close to postulate {such posits or postulated entities are myths from the standpoint of the level below them, the phenomenalistic level—[Douglas] Hofstadter} {kill or be killed the sergeants cried, discriminating Die from Live, and spoke the truth. And also lied, posited false alternative—[William] Gibson}
Evidently a sea change in the meanings of postulate and posit occurred between 1942 and 1984. In 1942 Webster's argues that postulate has an arm's-length relationship to its central assumption, while posit tends to embrace and believe in its assumption, to the point of treating it not as an assumption but as a point of faith. But in 1984 Merriam-Webster emphasizes the objectivity and rationality of postulate versus the subjectivity and arbitrariness of posit. The 1942 distinction focuses on the speaker's attitude toward the thing assumed, while the 1984 distinction seems to focus on the legitimacy or scientific defensibility of making the assumption.
I'm inclined to view Merriam-Webster's 1984 distinction between scientific postulating and capricious positing as an overstatement of the difference in how people actually use the two words. Certainly the mathematical prominence given to postulates in both Euclidean and non-Euclidean systems of geometry attaches a rigor to postulate that posit doesn't possess. But even if posit often indicates a speaker's rooting interest in the correctness of the asserted assumption, it needn't be altogether subjective and arbitrary. I confess that I don't see a particularly bright line separating Merriam-Webster's Huxley example ("the prevailing theological system is one which postulates the reality of guidance by a personal God") from its Santayana example ("if she needs salvation, she will posit a savior") with regard to objectivity/subjectivity or rationality/arbitrariness.
