I must put up a clarifying answer. They groped towards different concepts and talked past each other, in the modern perspective.
Much of what we call Newtonian mechanics is due to Euler, who called it such in his early papers on the topic, but Euler wrote after the controversy. (See Truesdell)
The problem is, as is clearly seen from the primary texts of Newton and Leibnitz, that Newton was working with the concept we call force (indeed, in the later edition of his optics, he discovered repulsive forces, in addition to attractive forces he postulated). This would lead to the progress made by Euler and Boscovich. Cartesius worked with momentum prior both.
Leibnitz was however concerned primarily with mechanics only so far as it concerned his "the most diverse one the one possible world that exists" philosophy. He argued that since all laws of nature can be written (truly modelled analogously) as equalities, the "whole effect" is always conserved. One side is cause and the other effect. This was intended to eliminate the concept and problem of "first cause" from an practical significance in his philosophical system of the world. Which side in an equation is the whole cause and which is the whole effect?
This is loosely similar to the modern position, because we no longer care or even assume existence of first causes, and it is not impossible that a time reverse situation occurs—for the cause and the effect in this case are in an equivalence class—but rather it is highly improbable, as determined by the thermodynamics of the situation. From here we have Boltzmann and Mach picking up the stick.
(The confusion early on was caused by the fact that modern theory in thermodynamics did not exist prior to Waterston, Mayer, and Boltzmann. Indeed, Leibnitz could not explain where the "effect" went when a body dropped from a height fell onto the floor, which ended its fall. That was the Mayer experiment 150 years later. And 15 years after than, even scientists such as Kelvin still had to be shown errors in their papers where they claimed they'd found violations of energy conservation—such as when Waterson pointed out to Kelvin that phase transformations do in fact require energy and cannot be ignored in well constructed experiments. The energy conservation concept was broadly spread at that point by the popular Spencer with his synthetic philosophy.)
Leibnitz simply extrapolated from a pendulum experiment that $mv^2$ is conserved in all cases. Such extrapolation is not logically kosher, true, but that was his hypothesis in his dynamics. He was essentially groping toward the modern energy concept, which was finally clarified by Mayer and Clausius and Mach and Einstein, in that order. It was Mayer who gave the first clear discussion of kinetic as distinguished from potential as distinguished from thermal energy. Maupertuis and D'Alembert and Euler worked on clarifying the concepts of kinetic and potential energy.
Without thermodynamics the controversy could not have been resolved given the state of knowledge when it occurred—and so indeed it wasn't. There are no miracles.