There are many notations for derivatives as the concept has been expanded in many different ways. For example, there is also Heaviside's operational D. This is also used for the Frechet and Gateaux derivative (which is implicitly used in the notation for tangent bundles in differential geometry).
Newton chose a notation for ease of use. As a physicist he was mostly interested in the first and second derivatives of time. Since the dependent variable implicitly understood, there is no need for the notation to reflect this. Hence he needed only to indicate the degree of the derivative. This is an integer. Buy since we are only interested in the first two, 1 & 2, we don't have to indicate the degree by a numerical prefix (as they do in some notations), we can simply indicate it by a single or double dot. It's quicker a d more convenient.
When one is interested in the calculus for its own sake, then a more comprehensive notation is necessary. This should indicate the degree, the dependent and independent variables. Thus Liebniz's notation is more natural here.
Had Newton been more interested in geometry than physics, and Liebniz more interested in physics than geometry it would have been likely we would have seen their notations being swapped. In other words, the notation associated with their names reflected their interests.
(It's worth adding that doing differential geometry with intuitionistic logic allows the introduction of infinitesimals that is much closer to how Newton envisaged them, his fluxions, rather than the traditional epsilon-delta techniques of traditional analysis. Moreover, they generalise to the infinite-dimensional context seamlessly, unlike the usual calculus for which there are many differing techniques).