My understanding has grown to be that Newton's genius was probably the incredible intersection of Calculus and Physics. (One or the other might have occurred to others, but Newton synthesized them better and faster and at the same time.) Newton's Calculus was rougher and aesthetically uglier, but he did the most the quickest with "Applied" Calculus. Leibniz had the raw math of the Calculus better and aesthetically his notation was much better. It was less "Applied" and didn't quite capture as much of the relationship to Physics (but it captured relationships to other parts of math that Newton missed, being so focused on Physics first).
We use Newton's insights into the applications of the Calculus to other sciences and we use Leibniz's artful way of capturing the Calculus to symbols on paper (and chalkboards).
There's a power of "dualism" in mathematics that sometimes you don't know that you have the right math until you've got two (or more views of it), enough to say "these lenses really do show the same thing".
I find it interesting to spot such dualisms, and especially how many of them were contemporary mathematicians working at some remove or another (countries or an ocean apart). From a computing perspective that's always been fascinating to me about the Church-Turing Theorem (which is always a relevant tangent on HN). Most people ignore Alonso Church's contributions and just call it the Turing Theorem, but the Theorem itself is about dualism (that all formulations we found of computation themselves are dual and can be translated or emulated between each other) and doesn't exist if it weren't for Church and Turing coexisting and conversing. (Plus, it has been said that it was Church that originally proposed the math to prove the dualism between their work and it was a sequence of correspondence that Church originated.)
That is no disrespect to Alan Turing, of course, to include Church in that theorem name and respect the dualism of their contemporaneous work. I think that balance looks a lot like Newton versus Leibniz: Turing knew the practical applications of computing and was starting from a place of having built them (though classified at the time) and Church was working from pure principles and theory in mathematics (the Lambda Calculus). We greatly benefit from both having worked on the same ideas as duals of each other, and we greatly benefit that their correspondence involved an ocean in between them so that they also weren't entirely in the same mindspace. We use a lot of Turing's applied practical synthesis of computing and we rely on a lot of things from Church's notations or things derived from it (including the Y Combinator for which this site's domain name refers).
Modern computing owes a lot to the dualism of Turing and Church. Modern Calculus owes a lot to the dualism of Newton and Leibniz. I find that incredibly interesting and I appreciate that about the mathematics of both things.