Does an Equivalence of Finitely Generated projective module categories over ring of functions imply a diffeomorphism between smooth manifolds?

Question

Let $M$ be a smooth connected manifold with $R = C^{\infty}(M,\mathbb{R})$ the $\mathbb{R}$-algebra of smooth real-valued functions on $M$. Let $ProjFinMod_R$ denote the category of finitely generated projective $R$-modules, and let $Vect(M)$ denote the category of smooth vector bundles of finite rank on $M$. The Serre-Swan Theorem describes an equivalence of categories $ProjFinMod_R \cong Vect(M)$. Given two smooth, connected manifolds $M,N$ with algebras of functions $R = C^{\infty}(M,\mathbb{R}),S = C^{\infty}(N,\mathbb{R})$ such that $ProjFinMod_R \cong ProjFinMod_S$, must $M$ and $N$ be diffeomorphic? If not, are they related through any sort of map?