What is are the benefits of map 500 hPa geopotential for weather forecasting?
The 500 hPa geopotential is usually called the Level of non divergence. It is understood that beneath that level there is a level of convergence and above that level there is a level of divergence so that this surface(500 hPa) is the surface to look for vertical motions. To  a first approximation one can assume the time derivative of density of a particular surface to be a constant and then set the continuity equation to be zero. 

$$\nabla. u = 0  $$

We can then expand on the terms on the LHS and separate the horizontal components from the vertical components. 

This turns out to be(in isobaric coordinates) 

$$ \nabla_H. u_H = -\frac{ \partial \omega}{\partial p}$$

One can then conclude from this equation the following

1) If the LHS is less than zero then there is an increase in vertical velocity with decreasing pressure

2) If the LHS is greater than zero then there is a decrease in vertical velocity with decreasing pressure

Since the atmosphere is bounded at two levels (the surface and tropopause) the vertical velocity is close to zero at these two levels. However in between these two boundaries the vertical velocity starts to increase and achieves it's maximum value before again tapering off at the tropopause. The level of maximum vertical motion is usually the 500 hPa and this is also called the level of non divergence.