Difference between smooth structures being equivalent or diffeomorphic.

Double check that my definition agrees with yours: a smooth structure $\mathcal{U}$ on a paracompact, second-countable, Hausdorff topological space $M$ is collection of chart neighborhoods $(U,\phi)$ such that:

• the $U$ cover $M$,
• the charts $\phi: U\to\Bbb{R}^n$ is a topological embedding, and
• for every pair $(U,\phi), (V,\psi)$ the transition map $\phi\psi^{-1}$ is a smooth map with everywhere-invertible differential from $\psi(U\cap V)$ to $\phi(U\cap V)$
• $\mathcal{U}$ is maximal with respect to these conditions, so that given another smooth structure $\mathcal{V}$, if $\mathcal{U}\cup\mathcal{V}$ is a smooth structure, then $\mathcal{V}\subset\mathcal{U}$.

I'm going to guess at the precise definitions of equivalent and diffeomorphic provided by your text.

We might say two smooth structures $\mathcal{U}, \mathcal{V}$ are diffeomorphic provided there exists a homeomorphism $h:M\to M$ such that the pullback of $\mathcal{U}$, defined as the structure $$h^*\mathcal{U} = \{(h^{-1}U, h^*\phi)\ |\ (U,\phi)\in\mathcal{U}\},$$ is equal to $\mathcal{V}$. Note that $h$ need not be smooth with respect to $\mathcal{U}$ or $\mathcal{V}$.

We might say two smooth structures are equivalent provided they are diffeomorphic by the identity map.