Differential or integral form of the conservation equations? Is there a 'rule' for when it is best to use either the differential or integral form of the continuity and momentum equations in calculations?
 A: I would say that you use integral forms of equations (the same is true in Electromagnetism) when there are high degrees of symmetry in the problem at hand. In the absence of symmetries, or if you are interested in what is going at a point or looking to introduce a numerical scheme for solving the physics, then usually the differential forms are best.
A: From a purely mathematical perspective, there really should not be any difference between the two forms. From a numerical simulation perspective, however, there can in fact be quite a bit of difference. In the application of the Navier-Stokes equations to Computational Fluid Dynamics (CFD) problems, it is often required to use the integral form due to the fact that the primary flow variables for which we are solving ($P,T,\rho,V,$  etc.) vary discontinuously across the domain (e.g. across a shock wave), and as a result make the gradient operators used in the differential form all but meaningless for computation. 
Secondly, it is an observed fact that conservation forms of the governing equations are generally more accurate[1], and tend to conserve mass better than the non-conservation forms when implemented correctly. The plot below shows a comparison between the two methods of calculating the nondimensional mass flow rate through a converging-diverging nozzle for fully subsonic flow.

[1] John Anderson, "Computational Fluid Dynamics," McGraw-Hill Education, Feb 1, 1995.
A: I managed to find an answer to my question:
"Frequently, we are interested in applying the basic laws to a finite region. Such equations are called global equations or simply integral forms of the equations." ~ Ronal L. Panton, Incompressible Flow.
It comes down to if the control region of interest is moving, the application of the integral form allows for this movement.
