What are the applications of hydrostatic solvers vs non-hydrostatic solvers in numerical weather prediction?
The hydrostatic approximation begins with the full 3-D momentum equation (Navier-Stokes) and through scale analysis the vertical momentum equation reduces to:

$$\dfrac{\partial p}{\partial z} = -\rho g$$

This is a balance between the vertical pressure gradient force and gravity with no net acceleration.  This tends to hold for atmospheric phenomena that are much wider than they are tall.  On the synoptic scale where vertical motions are on the order of 1 cm s$^{-1}$ this holds, but in a thunderstorm where updrafts can attain speeds of 100 m s$^{-1}$ this approximation doesn't work.

A hydrostatic model uses the hydrostatic approximation in place of an equation for $w$, while a non-hydrostatic model uses the momentum equation for $w$.  As the ECMWF  page you link to suggests, models with low grid resolution can get by with the hydrostatic approximation.  A model with dx=dy=10 km means the grid volume is 100 km$^{2}$ times whatever the vertical grid spacing is.  The $w$ value for that grid box needs to be representative of the entire volume and you aren't going to explicitly resolve convection at this resolution (convection will be parameterized and in general a model can only resolve features that are about 4x bigger than the grid spacing). 

As you reduce grid spacing (increase resolution) and start to resolve non-hydrostatic features (thunderstorm updrafts, downdrafts, etc), then you need to make sure you are using a non-hydrostatic solver.  For high resolution operational regional modeling and research modeling in the mesoscale using WRF-NMM, WRF-ARW, CM1, ARPS you'll find non-hydrostatic solvers.  

The non-hydrostatic solver is more computationally expensive than a hydrostatic solver, and is often found in higher resolution models which themselves are more computationally expensive than coarse grids.