In the Principia, after Newton introduces his definitions, and his scholium to the definitions, he turns the page and offers three "laws of motion." And while Newton's own formulation was displaced during the development of 'Newtonian' or 'classical' mechanics (Newton works with proportions not equalities), they have become familiar to us as "laws." But it is notable that Newton also introduces them as "Axioms."
I have never read a satisfactory account of this oddity, and here I want to start developing an answer.
Newton's use of 'Sive' is, I propose, like the Spinozistic 'sive' (as in Deus sive natura), where it really means, 'that is,' rather than a disjunctive 'or.' So, Newton is suggesting that one can treat the same entities as axioms and as laws of motion. Before I explain why he would suggest that, I first offer an interpretation of each on its own terms.
In the early modern period, an 'axiom' can be understood as an established principle in a science (see, for example, Chambers' Cyclopedia, 182). A 'principle," in turn, is a foundational, established ground or (general) cause. Newton is suggesting, thereby, that he is taking for granted a body of work (in 17th century mechanics). In fact, the three "laws" are, as Newton himself implies, implicit in and used by Huygens in his 1672 Horologium. To be sure, Newton's own way of presenting them, and interpreting their reach, was not standard; it was unprecedented to apply them to celestial affairs; nor was his way of understanding the third law of motion common (as became clear over its application to systems of bodies that are not in contact, but far apart).
That Newton treats them as a laws of motion is a firm nod to Descartes (as is widely recognized in the scholarly literature due to influence of I.B. Cohen and Howard Stein.) Newton's Principia is meant to displace Descartes' Principia -- with Newton proudly announcing that his are truly mathematical principles (with the implication being that Descartes only paid lip-serve to the role of mathematics in natural philosophy) -- and Newton's laws are meant to displace Descartes' purported laws.
But "laws" talk has a further implication: they are, in terminology of the age, second causes. That is, God is the first cause, but God's agency works by way of second causes (that is, laws of nature or laws of motion). Now, in Descartes scholarship it is an open debate what the status of laws are. I happen to agree with Hattab that for Descartes, laws really are genuine secondary causes. Similarly, I have argued that for Newton laws are really genuine causes -- akin (but not identical) to formal causes in the traditional, Scholastic conception, while forces are efficient causes for Newton. [To be sure there are alternative interpretations, but it is worth alerting the general reader that theses days few Newton scholars defend the once popular idea that Newton was an instrumentalist about either the laws or forces.]
So, by using 'Axioms, that is, laws of motion' Newton is signaling both (a) that he is developing his account from shared premises (this is also clear in his treatment of what counts as "phenomena" in Book 3 of the Principia), that's why he uses 'axioms,' as well as that these (b) may also be understood as a description of God's creation rather than a mere hypothetical model (as Descartes and Huygens tended to suggest); (b) fits Newton's language (he considers his own system as "true") as codified in his fourth rule of reasoning (but that was added only in third edition. As Andreas Hüttemann suggested to me in correspondence, (a) connects to the fact that Newton is offering a mathematical treatment (as 'axioms' are familiar from that use), while (b) reminds us that it is not a mere mathematical model, but identifying genuine causes.
As an aside, it is sometimes suggested (e.g., by Andrew Janiak and Steffen Ducheyne in their books on Newton) that Book 1 of the Principia is a mathematical treatment and Book 3 offers a physical (or causal) treatment; there is a way of reading Newton this way. But that has always struck me as misleading (not just because Book 2 ends up having an odd status). For, at the start of section 11 (of Book I), Newton acknowledges that the preceding results are all based on an un-physical assumption: They concern ‘‘bodies attracted toward an immovable center, such as, however, hardly exists in the natural world.’’ In light of the third law, Newton then introduces more realistic assumptions into his model that (in propositions 58–61) apply to Keplerian motion (of Earth–Moon–Sun systems). Subsequent sections relax even more of the simplifying assumptions, and the models become increasingly less idealized. This suggests that Newton recognized that his mathematical models had differing relationships to reality. This is especially evident in the scholia that Newton included in Book I— these often deal with issues of traditional natural philosophy; for example, the last two scholia and the surrounding propositions in the final section (14) of Book I treat of Snell’s law, the nature of lenses, and the speed of light. (For more on this, see my handbook paper with Chris Smeenk, or this review.)
I do not claim that the above is the final word on these matters. The true test for my interpretation lies in the nitty gritty detail of the wordings and applications of the 'laws.' For example, I have been silent here about one peculiarity of the laws of motion: they also hint at an account of body (although that is less complete than the treatment of motion). But that's for another time.