What is a "maximal collection"?
A differentiable structure is a collection of open neighborhoods and chart maps which satisfies a series of conditions:
- Open neighborhoods cover the manifold
- Open neighborhoods are homeomorphic to the model space $\mathbb{R}^n$
- On intersections of open neighborhoods, transition maps are smooth
To require maximality of a differentiable structure $\mathcal{A}$ is to require that if $\mathcal{A}$ is contained in any other differentiable structure $\mathcal{A}'$ which also satisfies these properties, then $\mathcal{A}' = \mathcal{A}$.
How to define a diffeomorphism between smooth manifolds?
A diffeomorphism between smooth manifolds is a homeomorphism that is smooth and smoothly invertible in every coordinate chart. Just as a homeomorphism defines an equivalence of topology between topological spaces, a diffeomorphism defines an equivalence of smooth structures.
So it looks like your job is to write down the maximal smooth structure $\mathcal{A}_1$ containing the identity map as a chart map, the maximal smooth structure $\mathcal{A}_2$ containing $t\mapsto t^3$, and then a self-homeomorphism of $\mathbb{R}$ (the underlying topological space) which is smooth and smoothly invertible in each coordinate chart in $\mathcal{A}_1$ and $\mathcal{A}_2$.