What is the difference between integral counting and differential counting?

I've never really heard those terms.

It makes me think of finding the area under a curve by adding up the area of an integral number of rectangles (a Riemann sum), and adding up the limit of rectangles of infinitesimal width (f(x)dx, where dx is the "differential"), which is the value of the integral.

I think it's getting as something like that.  You could find the distance a moving object travels by adding up a finite number of distances, like how far it goes in the first minute, second minute, etc.  Or you could add up the differential distance over infinitesimal units of time.

I'm sure you answer involves things like that.