Newton's method of successive approximation to roots of equations first appeared in print in 1685 in John Wallis's 'A Treatise of Algebra, both Historical and Practical' at pp.338-347, Chap.XCIV & XCV, 'A new method of extracting roots in Simple and Affected Equations' -- containing also Wallis's attribution to Newton. This presentation has been discussed by Niccolo Guicciardini, who has remarked that Newton's description 'is based on concrete examples ... a craft more than a theory': N Guicciardini (2009), 'Isaac Newton on Mathematical Certainty and Method'.
Newton published a further concrete example of the method in the 'Principia' (all editions, from 1687), at Book 1, Prop.31 (Scholium). This example was an application of the method, to solve an equation defining the orbital position of a planet, one version of Kepler's problem. The example was discussed in 1882 by John Couch Adams (Monthly Notices of the Royal Astronomical Society 43, pp.43-49), who pointed out that in the first edition of the Principia, Newton had tried to describe a refinement of the method, but had made a mistake (equivalent to forgetting to update one of the 'temporary variables' in the iteration); Newton corrected the description (as amended in the second and the third editions) with a simplified version of the quadratically-converging method. In the 1999 translation of Newton's Principia, the method appears at pp.514-515 (Isaac Newton, 'The Principia ... A new Translation', I Bernard Cohen, Anne Whitman, 1999).
Raphson's method of successive approximations to roots of equations was first published in his 'Analysis Aequationum Universalis', 1690, which went through at least three editions. (Neither Newton nor Raphson used any notation close to the modern " -f(x)/f'(x) " for the iterative corrections.)
Raphson's method has been discussed and compared with Newton's method by F Cajori among others (1911, American Mathematical Monthly, Vol.18, 29-32). Cajori showed how Raphson's method resembled that of Newton especially in that they both used a divisor for the next correction that evaluates, in effect, to (what in modern notation would be) f'(r), where r was the most recent corrected value. (But neither of them used a symbolic notation such as f'(r), they expressed each case of the quantity in an extended e.g. polynomial form.)
But where Newton had derived each successive step of approach to the root from a newly formed equation derived from the original one, Raphson, closer in that respect to the modern procedure, found each successive corrected value by substitution in the original equation. (Cajori also remarked that Newton's forerunners such as Vieta had used a different and generally non-equivalent principle for evaluating the relevant divisor. But in some specific example-cases, Vieta's value of the divisor had coincided with the value of Newton's divisor -- which had led some commentators mistakenly to assimilate Newton's method with Vieta's.)
Cajori gave a history of the methods, and in view of Raphson's important modification proposed the name 'Newton-Raphson' method as a more accurate attributive label than just the name 'Newton's method', which had been current. His suggestion, made over a century ago now, does seem to have taken hold, 'Newton-Raphson' is widely used to denote the modern version of the method.