Why work with $C^\infty$ manifolds as opposed to $C^k$ manifolds for $k \in \{1,2,...\} \cup \{\infty\}$?

Question

In a book like Tu, Introduction to Manifolds, only smooth (here meaning $C^\infty$) manifolds are considered. Once we have any differentiable structure, i.e. we have a $C^k$ manifold for some $k \in \overline{\mathbb{N}} := \{1,2,...\} \cup \{\infty\}$, then we get nice things like the tangent space functor. What properties/theorems hold for smooth manifolds that don't hold for $C^k$ manifolds for all $k \in \overline{\mathbb{N}}$? What advantages are there to working with the seemingly "nicer" smooth manifolds? What are some examples of proofs that wouldn't go through with every instance of "smooth" replaced with "$C^k$ for any $k \in \overline{\mathbb{N}}$"?