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.
