What's the difference between the two equations for work?
Aside from a sign convention artifact, they're the same thing.

Pressure is force per unit area:

$$
 P = {F\over A}
$$

If a surface with a given area is moved through a displacement $\Delta x$, then the volume of displacement is the product of $A$ and $\Delta x$:

$$
\Delta V = A \Delta x
$$

So, the $P\Delta V$ work formula can be rewritten:

$$
w = -P\Delta V = -P\left(A \Delta x\right) =
  - \left({F \over A}\right)\left(A \Delta x\right) =
  - F \Delta x
$$

There are a lot of assumptions embedded in the above, and you have to be a lot more careful than I was about defining positive and negative quantities correctly, but on a high level that's all there is to it.