Polynomials can be used to extract information about finite sequences much in the same way as generating functions can be used for infinite sequences. However, it is good to note that generating functions are not always more suitable for such purposes than polynomials; polynomials allow more operations and convergence issues can be neglected. In fact, one does even need the underlying field to have any topological structure.

Let me give some examples where polynomials are more appropriate than generating functions. For example, the combinatorial nullstellensatz is a useful tool in combinatorics, particularly in additive problems, and its formulation is based on polynomials over a field. Another important example is that one obtains better results in the Hardy-Littlewood circle method by replacing infinite Fourier series with trigonometric polynomials. Third exmple is that manipulating suitable polynomials in $\mathbb{Z}/p\mathbb{Z}$ gives some interesting reults in number thory, such as Wolstenholme's theorem. Fourth example is that irrationality and transcendence proofs are usually based on considering suitable polynomials (for example, in the case of $\pi$).

In addition, polynomials sometimes serve as ''generalized integers'' in number theoretical contexts. Many theorems are easier to prove for them (such as Fermat's last theorem for non-constant polynomials), and they can be used to conjecture results about integers; this is how abc conjecture was found, for instance.

On the other hand, in complex analysis, polynomials are basic examples instead of generalizations. There are numerous results in complex analysis that are easy for polynomials but generalize in an interesting way to analytic functions, and on the other hand, there are also many results where the polynomials are the only exceptions due to their slow growth or rigidness. The fact that polynomials are determined by their values in finitely many points is indeed another reason for their usefulness in analysis and other branches as well.