A celebrated theorem of Whitney says that every $C^1$-manifold can be made a smooth manifold by considering an appropriate differentiable structure.
On the other hand, a set $M\subseteq \mathbb R^n$ (with the subspace topology) has the property of being a $C^k$-manifold means that, for every $x\in M$, there are an open set $V\subseteq \mathbb R^m$ containing $x$ and an injective $C^k$-function $f:U\to\mathbb R^m$ on some open set $U\subseteq\mathbb R^n$ such that $M\cap V=f(U)$, $f^{-1}:M\cap V\to U$ is continuous, and $f'(t):\mathbb R^n\to\mathbb R^m$ is injective for all $t\in U$.
I am quite sure that Whitney's theorem does not say that the properties of being a $C^k$- or a $C^p$-manifold are equivalent for different $k,p\in\mathbb N$.
What are simple examples?
If $g:\mathbb R \to \mathbb R$ is $C^1$ but not $C^2$ (e.g., $g(t)=0$ for $t\le 0$ and $g(t)=t^2$ for $t>0$), the graph $\{(t,g(t)):t\in\mathbb R\}$ is clearly a $C^1$-manifold. I am very embarrassed that I don't see if it can be $C^2$.
The curve you mentioned cannot be $C^2$ because its curvature function is discontinuous at the origin. If you approach the origin from the left, the curvature is zero. If you approach from the right, the curvature tends to 2.