Why is it important to have the closed form of a generating function?
There are many reasons. Here are some:
Closed forms facilitate calculations with generating functions. Certain combinatorial operations correspond to simple operations on the generating functions such as addition and multiplication, and these are facilitated by working with closed forms.
Proving identities. Continuing the previous bullet, manipulations of generating functions leading to the zero function correspond to a combinatorial identity stating that matching coefficients are equal. This is important for automated methods for proving combinatorial identities, such as the ones described in the book A=B. Other examples are proofs of certain identities regarding the partition function, and one proof of the four square theorem.
Extracting coefficients. As you mention, if you just take a given sequence and transform it into a generating function, this seems pointless. But sometimes generating functions can result from solving equations – the classical example is the Fibonacci sequence. Or they could result from manipulations as described in the first bullet. There are techniques, such as partial fraction decomposition, to extract coefficients from generating functions, and they can be used to find formulas for recurrence sequences.
Asymptotic analysis. Probably the most important reason to use generating functions is that they allow the use of analytic methods to determine the order of growth of the coefficients, using various techniques. An offshoot of this approach is the method of characteristic functions in probability theory.