New proofs to major theorems leading to new insights and results?
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.
A very nice example in my eyes is Serre's proof of Riemann-Roch:
Sometimes, you are just not satisfied with existing proofs, and you look for better ones, which can be applied in different situations. A typical example for me was when I worked on the Riemann-Roch theorem (circa 1953), which I viewed as an "Euler-Poincare" formula (I did not know then that Kodaira-Spencer had had the same idea.) My first objective was to prove it for algebraic curves - a case which was known for about a century! But I wanted a proof in a special style; and when I managed to find it, I remember it did not take me more than a minute or two to go from there to the 2-dimensional case (which had just been done by Kodaira).
He is speaking, of course, of the sheaf-theoretic proofs, which are usually presented today. This was the period where he was working on FAC, GAGA and his duality theorem, which revolutionized algebraic geometry.
The classic example from mathematical physics is Richard Feynman's Space-Time approach to nonrelativistic quantum mechanics (1948), which (in essence) proved that the Green function of the Schroedinger equation was equal to a path integral. The article begins:
It is a curious historical fact that modern quantum mechanics began with two quite different mathematical formulations: the differential equation of Schroedinger, and the matrix algebra of Heisenbert. [...] This paper will describe what is essentially a third formulation of non-relativistic quantum theory.
As for the value of seeking multiple derivations, we have Feynman's Nobel Address The Development of the Space-Time View of Quantum Electrodynamics (1965):
There is always another way to say the same thing that doesn't look at all like the way you said it before. I don't know what the reason for this is. I think it is somehow a representation of the simplicity of nature. [...] Perhaps a thing is simple if you can describe it fully in several different ways without immediately knowing that you are describing the same thing.
In a classical context, we have Saunders Mac Lane in Hamiltonian mechanics and geometry (1970) presenting new geometric analyses of old dynamical problems:
Mathematical ideas do not live fully till they are presented clearly, and we never quite achieve that ultimate clarity. Just as each generation of historians must analyse the past again, so in the exact sciences we must in each period take up the renewed struggle to present as clearly as we can the underlying ideas of mathematics.
In the mid-1970s these various derivations came together as Fadeev and Popov's (1974) Covariant quantization of the gravitational field, which provided the foundations for todays' gold-standard method of BRST quantization, for which van Holten's Aspects of BRST quantization (2002) is a good review:
Quite often the preferred dynamical equations of a physical system are not formulated directly in terms of observable degrees of freedom, but in terms of more primitive quantities [...] Out of these roots has grown an elegant and powerful framework for dealing with quite general classes of constrained systems using ideas borrowed from algebraic geometry.
By this 90-year process of successive rederivations, we nowadays have arrived at a more nearly global appreciation—encompassing both classical and quantum dynamics—of the ideas that Terry Tao's essay What is a Gauge? discusses.
Cutting-edge research in classical, quantum, and (increasingly common) hybrid dynamical systems uses all of these mathematical approaches, each formally equivalent to all the others ... but with very different ideas behind them. The resulting naturality has lent new passion to the longstanding romance between mathematics and physics.