electromagnetism - What are the differences between the differential and integral forms of (e.g. Maxwell's) equations?

I would like to understand what has to be differential and integral form of the same function, for example the famous equations of James Clerk Maxwell:

How to know where to apply each way? Excuse the ignorance, but always confused me the head.

Edit

Remembering that, I know that the concept of integration and derivation, is something like:

Answer

The equations are entirely equivalent, as can be proven using Gauss' and Stokes' theorems.

The integral forms are most useful when dealing with macroscopic problems with high degrees of symmetry (e.g. spherical or axial symmetry; or, following on from comments below, a line/surface integrals where the field is either parallel or perpendicular to the line/surface element).

The differential forms are strictly local - they deal with charge and current densities and fields at a point in space and time. The differential forms are far easier to manipulate when dealing with electromagnetic waves; they make it far easier to show that Maxwell's equations can be written in a covariant form, compatible with special relativity; and far easier to put into a computer to do numerical electromagnetism calculations.

I would think that these three points generalise to any system of differential vs integral forms in physics.