The differential forms of Maxwell's equations, like Gauss's Law help tell us how a field behaves at a point, which the integral forms cannot tell us about. Other than this one difference, they describe the same physical phenomena. Whether you use one form or another depends on how useful that form is to the problem your working on.
The divergence is just a vector derivative:
[tex]\nabla\cdot\vec{v}= \frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}[/tex] In fact, just as there are two ways to multiply vectors (dot and cross products), there are two ways to differentiate them. You can take a vector's divergence, or it's curl. Both are derivatives, but they tell you different things. The divergence tells you how much the vector field diverges from a point, i.e. the electric field from a positive point charge had a high divergence. (It always points away from the source of the field. In other words, it diverges from that point.) On the other hand, the magnetic field of an infinite current-carrying wire, which loops around the wire has zero divergence (the field lines are always the same distance from the wire), but they have a high curl (the field lines "curl" back on themselves.)