First, note that $\mathcal{C}^{\infty}$ is a soft sheaf, hence acyclic.
It follows that $H^1(M,(\mathcal{C}^{\infty})^{\times}) \cong H^2(M,\mathbb{Z})$ for any smooth manifold $M$. In particular, if $M$ is compact, there are only countably many isomorphism classes of smooth line bundles.
Now assume that $M$ is a compact Riemann surface of nonzero genus. Then its Jacobian $J$ parametrizes a subset of isomorphic classes of line bundles on $M$, and $J$ has uncountably many points – so some of them correspond to line bundles that are smoothly isomorphic.
(Using more carefully the ideas of First Chern class for smooth line bundle , we should be able to prove that every holomorphic line bundle of degree zero on a Riemann surface is smoothly trivial).