From a conversation I had with Gian-Carlo Rota when I was undergraduate, I know that one simple but important example that he specifically had in mind was the calculus of vector fields (whether specifically in three dimensions or more generally). The gradient, divergence, and curl of differentiable fields on ${\mathbb R}^{3}$ can be defined as particular combinations of partial derivatives—in which case it is necessary to prove that they represent geometrical objects (meaning they transform correctly). Alternatively, it is possible to specific purely geometrical definitions of all three objects, in which case it is necessary to prove that, when applied to sufficiently smooth functions, the can be calculated entirely in terms of partial derivatives. Whichever way you like to approach the theory, it is possible to find textbooks that take your preferred starting point and do a good job of explaining vector calculus—even though the two approaches are, philosophically, quite different in terms of what they seem to assume about what, say, $\operatorname{grad} f$ "really means." Moreover, there are also plenty of important theorems that can be proven from either starting point, without proving the equivalence first.
Somebody else, in the course of that conversation, mentioned the logarithm as an even more basic example. There are actually many ways of initially defining the logarithm, and Calculus by James Stewart (or the first edition, at least) actually demonstrates explicitly that you can begin with the logarithm as the inverse of the exponential, or you can define $\ln x=\int_{1}^{x}(1/t)\,dt$ and eventually prove all the same things.
