On a manifold equipped with C^k atlas (with k>0) there is essentially one smooth structure compatible with the atlas. According to Wikipedia, this is a result due to Whitney. This is in stark contrast with a C^0 atlas, where there might exist many smooth structures or none at all.

I was wondering, what is the underlying reason? What makes once-differentiable functions so much better behaved in terms of finding a smooth atlas?

There are many cases where C^1 makes a world of difference - for example, convergence of Fourier series, but maybe there is some geometric explanation?

Also according to Wikipedia, on the long line (not technically a manifold) there are infinitely many smooth structures all compatible with a given C^k structure, so perhaps there is some topology involved...