What is hydraulic diffusivity?
So let's get through some definitions. I will not discuss the derivations of this, but you can look this up, if you want to in the source I provided.

We find ourselves in a porous medium, so we will have always some volume filling factor $\Theta$ of water in rock.

 (copyright K. Roth, Heidelberg University)

Then we can start with the hydraulic conductivity, that can be defined as $$K = \frac{\Theta \left&lt; \kappa \nabla P \right&gt;}{\mu}$$
Here we have quite some notation to clarify:


$\Theta$ is the local filling fraction of water in an aquifer
$\mu$ is the molecular diffusivity
The brackets $\left&lt; ... \right&gt;$ denote the macroscopic average over the microscopic aquifer structure that is given by Permeability $\kappa$ and local pressure gradient.
The Permeability itself is a way to encapsulate how easy water can flow through a given geometry.


This is the most rigorous definition I could find in my notes. The lecture however is unavailable on the internet, so you'll have to take my word for this. I just mention this as I don't know whether you're more interested on the theoretical side, or the pragmatic.

The pragmatic approach is usually to simply state Darcy's law $$ q = - K \frac{dh}{dz} $$
relating the water mass flux $q = \rho v$ to the gradient of hydraulic head.
We see now that a flow velocity will result from a pressure gradient, but strongly modified through the microscopic properties encapsulated in K.



The hydraulic diffusivity you search for is then written as $$D = \frac{K}{S}$$ where S is the water storage fraction, with $S \sim \Theta$ (not strictly correct). So in a way this just cancels the $\Theta$ in $K$ and appears in the pressure diffusion equation:
$$ \partial_t P = D \Delta P $$
and therefore D gives the local diffusion speed of pressure disturbances (which is interestingly not the speed of sound $c_s$ in aquifers!)

We could now say, for educatory purposes that the difference between $K$ and $D$ is only $\Theta$. This makes sense when we look at the context where those two parameters appear:
$K$ gives the strength of mass-flow for the fluid, the better filled the volume is, the more mass will flow.

Pressure, however does not care for the amount of water that flows, as it exists everywhere where water is sitting at a given moment, so the coefficient for its propagation does not depend on the filling of volume.

You can look up the derivations for example here.
Tell me in the comments, if anything is unclear.