What is the significance of polar differentiation and integration?

I think it is less a question of significance and more a question of convenience.  The cartesian coordinate system defines points in a very triangular way.  In essence you triangulate the position of each point.  For some functions this works really well.  For more circular and by extension spherical equations defining the functions differently the polar coordinates are easier to work with.  It is not a question of them being more or less useful, it is a question of the functions being easier to manipulate in polar.  
If you think about it, the slope in the cartesian plane  is dy/dx, change in vertical to change in horizontal.  In polar it is usually dr/d(theta), so it is the change in the radius from the origin as the angulation changes.  This would tell you something about the shape of the function you are looking at.  For example, a circle has constant radius, so dr/d(theta)=0.  Whereas an oval would change for all theta