I am wondering, historically, when has a new proof of an old theorem been particularly fruitful. A few examples I have in mind (all number theoretic) are:
First example is classical… which is Euler's proof of Euclid's theorem which asserts that there exist infinitely many primes. Here is when the factorization $\displaystyle \prod_p (1-p^{-s})^{-1} = \sum_{n=1}^\infty \frac{1}{n^s}$ was first introduced, leading of course to what is now known as the Riemann Hypothesis.
Second example is when Hardy and Littlewood gave an alternative proof of Waring's problem, which was done by Hilbert earlier. Their proof introduced what is now known as the Hardy-Littlewood Circle Method and gave an exact asymptotic for the Waring bases, which is stronger than Hilbert's result which only asserted that every sufficiently large positive integer can be written as the sum of a bounded number of $k$th powers. Later on the Hardy-Littlewood method proved very fruitful in other results, namely Vinogradov's Theorem asserting that every sufficiently large odd positive integer can be written as the sum of three primes.
Third example is Tim Gowers' alternate proof to Szemerédi's Theorem asserting that every subset of the positive integers with positive upper density contains arbitrarily long arithmetic progressions. This advance, namely the introduction of Gowers uniformity norms, led eventually to the Green-Tao Theorem proving the existence of arbitrarily long arithmetic progressions in the primes.
So I am wondering if there exist other incidences (number theory related or not) where a new proof really gave legitimate new insights, perhaps even a proof of a (major) new result.
Edit: I am primarily interested in examples where a new proof sparked off a new direction in research. This is best supported by having a major new theorem proved using techniques inspired by the new proof. An example of something that I am not interested in is something like Donald Newman's proof of the prime number theorem, which while elegant and 'natural' as he puts it, has seen limited generalization to other areas and one is hard pressed to apply the same technique to other problems.
Best Answer
Here are a few examples from the 19th century.
Unsolvability of the quintic equation. Abel (1826) proved this by algebraic ingenuity, but without clarifying the concepts involved. Galois (1830) gave a proof that introduced the concepts of group, normal subgroup, and solvability (of groups), thus laying the foundations of group theory and Galois theory.
Double periodicity of elliptic functions. Abel and Jacobi established this (1820s) mainly by computation. Riemann (1850s) put elliptic functions on a clear conceptual basis by showing that the underlying elliptic curve is a torus, and that the periods correspond to independent loops on the torus.
Riemann-Roch theorem. Riemann (1857) discovered this theorem using Riemann surfaces, but applying physical intuition (the "Dirichlet principle"). This principle was not made rigorous until 1901. In the meantime, Dedekind and Weber (1882) gave the first rigorous and complete proof of Riemann-Roch, by reconstructing the theory of Riemann surfaces algebraically. In the process they paved the way for modern algebraic geometry.