Are the equations of motion more fundamental than the conservation of energy in mechanics?

In some cases, it is possible to show the conservation of energy by using equations of motion in mechanics. By considering these results, some people can think that the conservation of energy is the result of equations of motion or Newton's second law. If we consider the conservation of energy by itself, it is valid for nearly all natural sciences and more general than equations of motion. From this perspective, it is not totally convenient to say that the conservation of energy is the result of equations of motion. It is clear that there can be a relation between them, but it is not explicit enough. In this study, we have studied the relationship between the conservation of energy and equations of motion. And, the study revealed a subtle difference between them in mechanics which can be used to experimentally test which explains the nature best or to better understand the relation between them.