Feynman tries to account for the relation between the principle of stationary action, which is a statement about the whole path of a particle, and Newton's second law, which is a statement about the instantaneous state of a particle. In the paragraphs, he means by the "differential" law, Newton's second law.
Now, I would like to explain why it is true that there are differential laws when there is a least action principle of this kind. The reason is the following: Consider the actual path in space and time. As before, let’s take only one dimension, so we can plot the graph of x as a function of t. Along the true path, S is a minimum. Let’s suppose that we have the true path and that it goes through some point a in space and time, and also through another nearby point b (Fig. 19–11). Now if the entire integral from t1 to t2 is a minimum, it is also necessary that the integral along the little section from a to b is also a minimum. It can’t be that the part from a to b is a little bit more. Otherwise you could just fiddle with just that piece of the path and make the whole integral a little lower.
“So every subsection of the path must also be a minimum. And this is true no matter how short the subsection. Therefore, the principle that the whole path gives a minimum can be stated also by saying that an infinitesimal section of path also has a curve such that it has a minimum action. Now if we take a short enough section of path—between two points a and b very close together—how the potential varies from one place to another far away is not the important thing, because you are staying almost in the same place over the whole little piece of the path. The only thing that you have to discuss is the first-order change in the potential. The answer can only depend on the derivative of the potential and not on the potential everywhere. So the statement about the gross property of the whole path becomes a statement of what happens for a short section of the path—a differential statement. And this differential statement only involves the derivatives of the potential, that is, the force at a point. That’s the qualitative explanation of the relation between the gross law and the differential law.
Given a path whose action is minimum, it follows that every arbitrarily small subsection of this path has minimum action as well. I accept this deduction.
Feynman then tells us, to consider such an infinitesimally small subsection of the whole path.
The part that I don't get: He says that the change in potential energy is not important, but rather the first order change in it, or in other words, the derivative of potential energy with respect to position, that is the force.
I know that force is given by the derivative of the potential energy, and that the first order change in the potential energy is given by the derivative. However I can't get why it's only the first order change that matters. Also, how an infinitesimally small subsection of the whole path having minimum action implies we should consider the first order change in the potential energy?