A good guess, but as it turns out neither equation comes from an underlying differential equation. In fact they are both reflective of ideas from equilibrium statistical mechanics, in particular the Boltzmann distribution, which tells you that the relative probability of observing a system in state s2 versus state s1 P2/P1 = exp(-(E2-E1)/kT) where E1 and E2 are the energies, T is the temperature, and k is the Boltzmann constant, whcih is just R divided by the Avogadro number.
You can probably see immediately how this applies to the vapor pressure case: the two states are the molecule in the gas phase and the molecule in the condensed phase, and the difference in energy between the two states is just the enthalpy of vaporization.
The Arrhenius equation is trickier, because you'd think it *would* be derived from a differential equation. But it's not. It's derived by assuming a very simple model of chemical kinetics: that the rate of a chemical reaction depends on the probability that on any given collision the collision energy exceeds the activation energy Ea. Clearly that probability, at equilibrium, is again given by the Boltzmann distribution, only this time the energy difference is the activation energy. There is a (very complicated) prefactor in the Arrhenius equation that takes into account genuine kinetic factors, such as the rate of collisions. And if you want to get into genuine chemical dynamics factors, which include the possibility that the reaction may NOT proceed to completion, just because the energy is above the barrier, then you are no longer working in the simple Arrhenius model.
There are many more examples of equilibrium and pseudo-equilibrium laws in chemistry that have their roots in the Boltzmann distribution as well. One of the most obvious is the fact that the pressure of the atmosphere falls off exponentially with altitude.