When differential and integral calculus were first discovered, in the 1600-1700s, they were proven to be immensely useful in so many applications it is almost mind boggling. But as far as I know, it was first long into the 1900s before any strict theoretic foundation for infinitesimals was actually established.
In many modern physics and engineering applications, concepts such as multiresolution analysis, scale spaces and coarse-to-fine play an important role. Roughly speaking in those concepts short time (or length) integrals and differential operators are mixed together. Which size portion of each decides the scale / resolution.
Is there some connection between how infinitesimals were first formally defined and these developments?
If multiresolution analysis, scale spaces, or coarse-to-fine methods are branches of applied mathematics (in which I am not an expert) then there is a good chance Abraham Robinson may have had something to say about them, because Robinson was an applied mathematician and in fact published a book called Wing Theory in the 1950s. In fact his book introducing infinitesimal analysis contains several applications in applied mathematics; see this MO post.