The integral formula is a lot of sums of very small lines (you know what I mean). The vectors representing those lines talk about how quickly the function's value is changing. So if you sum up the lengths of those vectors, you get the large scale change of the function.
The reason, then, that the two formulas look so similar is because those vector's lengths were needed to compute the components of the arc length.
I highly doubt there is any deep reason one tends to be line integrals and the other surface, but keep in mind that one is measuring things about a curve and the other could find something about a vector normal to the surface, which as you know from linear surfaces tend to reveal a lot about the surface itself.