I am not sure if this question is proper, if not please let me know I would delete it.
I understand a function can be continuous but not differentiable. I can also understand existence of non-differentiable functions. The question: is there a 'measure' (like distance) between two non-differnetiable functions. The word 'measure' here should not be taken literally. Consider two non-differentiable function $f(x)$ and $g(x) $ the question: is there a way to compare the two functions (like a distance measure ) which tells either $f(x)$ or $g(x)$ is easier to make it differentiable? Something like a how far the function is away from differentiable?
There is Rademacher's theorem which says that a function that is Lipschitz continuous is differentiable (almost everywhere). So one way of interpreting your question is to "how close to Lipschitz continuous is a function". Lipschitz continuity is a specific case of Hölder continuity.
A function is $\alpha$-Hölder continuous for $\alpha\in (0,1]$ if
$$|f(x)-f(y)|<K|x-y|^\alpha$$
If $\alpha=1$ then we say the function is Lipschitz.
Note that if $f$ is $\alpha$-Hölder, then it is $\beta$-Hölder for $\beta<\alpha$.
So if $f$ is $.6$-Hölder and $g$ is $.7$-Hölder, then $g$ is "closer" to being Lipschitz, i.e. differentiable.
Here are some pictures of various functions that are of increasing Hölder continuity:
Hölder continuous for all $\alpha \in(0,0.15)$:
Hölder continuous for all $\alpha \in (0,0.55)$:
Hölder continuous for all $\alpha\in (0,0.75)$:
Hölder continuous for all $\alpha\in (0,0.95)$:
Note how the paths get "nicer"/closer to being able to differentiate.