Difference between $C^k$s and $C^\infty$ What is the true difference between $C^0$, $C^1$, $\dots$, and $C^\infty$ objects and morphisms? (Or less dramatically formulated, what are deep differences between them?)
For instance, we do know that the existence of certain maps of classes $C^0$ and $C^\infty$ on $C^\infty$-manifolds is equivalent, as $C^0$ maps can be arbitrarly approximated by those of class $C^\infty$, even in a manner that respects $C^\infty$-ly treated closed subsets.
The lists of connected closed surfaces essentially coincide in $C^0$ and $C^\infty$. (That is, exactly one smooth structure per genus and orientability.)
There is exactly one $C^\infty$-structure on the topological space $S^2$.
Not every $C^k$-manifold is $C^\infty$, as we might consider an embedded cube in $\mathbb R^n$ with edges that are $C^k$, but not $C^{k+1}$.
Are there possibilities to classify all $C^k$-structures on a given topological space, say for $k\ge 1$, just by knowing the nature of its smooth structures, in some more general setting than those of closed surfaces?
 A: There are huge differences between $C^0$ and $C^1$; from the perspective of manifolds and maps between them, there is not much difference between any of the $C^k$ with $1 \leq k \leq \infty$. I will be precise about how.
1) Every $C^k$ atlas on a manifold $M$ has a subordinate $C^\infty$ atlas. Given a $C^k$ structure on a manifold $M$, any two compatible smooth structures on $M$ are $C^\infty$ diffeomorphic. So from the perspective of classification of manifolds, there is no difference between $C^k$ and $C^\infty$, $k>0$.
2) The spaces of maps $C^k(M,N)$ (take $M$ to be compact and the Whitney topology) are homotopy equivalent for all $k$. This is more or less because of the smooth approximation theorem.
3) For $\infty \geq \ell \geq k > 0$, the inclusion $\text{Diff}^\ell(M) \hookrightarrow \text{Diff}^k(M)$ is a homotopy equivalence. This also follows from smooth approximation theorems; the point is that $\text{Diff}^k \cap C^\ell = \text{Diff}^\ell$ is open in $C^\ell$. This is the strongest form of 1) (especially if you replace this by $\text{Diff}^k(M,N)$); not only does it say that $C^\ell$ manifolds $M$ and $N$ are $C^k$ diffeomorphic iff they're $C^\ell$ diffeomorphic, it says that their spaces of symmetries are homotopy equivalent (which in my eyes means 'pretty much the same'!) 
