The Atiyah-Singer index theorem has many different proofs: by cobordism theory and K-theory, by the heat equation method, and proofs coming from mathematical physics. I guess that there are further distinct proofs.

I am aware of two diffrent proofs of the theorem that Riemannian manifolds of constant curvature are locally isometric to spherical, euclidean or hyperbolic space: the first proof uses Jacobi fields, while the second uses Frobenius theorem.