What is the significance of this theorem (coefficients of power series as integrals)? 
Isn't it easier to obtain the series' coefficicients by differentiation rather than by integration?
The above text uses this theorem as an intermediate step in obtaining the generalised Cauchy integral formula, so I'm guessing that this theorem is useful mainly as a lemma?
On the other hand, another text I'm reading obtains the generalised Cauchy integral formula by simply "keeping track of the exponents", intead of relying on the above theorem.
 A: It's true that in practice one rarely obtains the coefficients by actually doing the integration.
However, you can use this to get useful estimates on the coefficients.  But the main significance of the theorem is that it means that the "analytic" functions (i.e. those defined locally by convergent Taylor series) and the "holomorphic" functions (i.e. those differentiable in the complex sense) are the same.
A: The Taylor coefficients of a function can of course be calculated by differentiation, but the whole point is that we can instead obtain them by integration. Intuitively, differentiation makes things more "jagged", while integration makes things more "smooth", and the Cauchy integral formula tells you that in fact complex differentiation on holomorphic functions does not make things more jagged, which is one of the reasons why the study of complex functions is so much nicer and elegant than the study of real functions.
A: The main application of this theorem is in getting estimates for coefficients growth in terms of the growth of the function.
For example one can directly get The Liouville theorem (http://goo.gl/sN5JT)
from the Cauchy formula
