I am already aware of the notation differences. But is it the only criterion?
Based on written/published records.
First written record from Newton: De analysi per aequationes numero terminorum infinitas (written: 1669; published: 1711).
First published record from Newton: Tractatus de Quadratura Curvarum (1704).
First published paper from Leibniz: Nova Methodus pro Maximis et Minimis (1684).
I will admit to the breaking of a rule in this case and offering an opinion because we are referring to a controversy that set England and Continental Europe against each other, only matched in personal ferocity by the argument between Jacob and Johann Bernoulli about which of them first successfully applied the Calculus (which they learned through correspondence with Leibniz) to derive the function that accurately described the catenary curve (how suspension bridges and necklaces hang). Opinion is really all we have.
The Two crucial points would seem to be--in my view: 1) His introduction of differential notation. 2) His principle of the Identity of Indiscernibles, which implies a (finite) limit of discernibility. The two together indicate that long before Weierstrass (1872) had to point it out, commencing the Great Crisis of Mathematics and the separation of intuitionism from realism, Liebnitz understood that differential calculus was about an ultimately fine-grained finite algebra--not actually the infinitely small. Assuming this superior insight, he had a truer grasp of what the Calculus was about than Newton.
[The universe is quantized after all, and not infinitely divisible. Thus, differential and integral calculus are not intuitionist infinite step constructs but approximations to the finite step constructs of science and engineering, and actually, work.]