In the theory of electromagnetism, the classical Stokes Theorem moves between the differential and integral forms of two of Maxwell's four equations; see https://en.wikipedia.org/wiki/Stokes%27_theorem#In_electromagnetism for discussion. Note that the integral forms may be directly motivated by interpreted using classical physical intuition, while the differential forms give us differential equations that we might solve, so it is important that we can switch between them.
ETA: I think that Wikipedia's discussion is a little vague, although possibly appropriate in that context. So here is more detail, looking at Faraday's Law. In terms of physically observable quantities, the law states that the rate of change of the magnetic flux through a stationary surface is proportional to the electromotive force around the boundary of the surface. The magnetic flux is the surface integral of the magnetic field $ \vec H $, and the EMF is the line integral of the electric field $ \vec E $, so we have $$ \oint _ { \partial S } \vec E \cdot \mathrm d \vec r = - \frac { \mathrm d } { \mathrm d t } \iint _ S \vec H \cdot \mathrm d ^ 2 \vec A $$ using standard units and sign conventions. Applying the classical Stokes Theorem on the left and using that $ S $ is stationary on the right, this becomes $$ \iint _ S ( \nabla \times \vec E ) \cdot \mathrm d ^ 2 \vec A = - \iint _ S \frac { \partial \vec H } { \partial t } \cdot \mathrm d ^ 2 \vec A \text ; $$ since this holds for arbitrarily small surfaces, we conclude that $$ \nabla \times \vec E = - \frac { \partial \vec H } { \partial t } \text , $$ a differential equation. (The argument in reverse is even easier, since you don't have to worry about arbitrarily small surfaces.)
