Number of Differentiable Structures on a Smooth Manifold

The distinction to be made is that a differentiable structure is a choice of maximal smooth atlas $\mathcal A$, but two different choices $\mathcal A$ and $\mathcal A'$ can lead to isomorphic smooth structures. As an example, the canonical smooth structure $\mathcal A$ on $\mathbb R$ that contains the smooth function ${\rm id}:\mathbb R\longrightarrow \mathbb R$ is isomorphic to the smooth structure $\mathcal A'$ that contains the smooth function $x\mapsto x^3$, although $\mathcal A'\neq \mathcal A$. Thus, although a manifold admits uncountably many different smooth structures, it may have finitely many isomorphism classes of such structures.

In the second statement, "unique" means unique up to diffeomorphism.

If you have a manifold $M$ with a smooth structure $A$ and a homeomorphism $\varphi :M \rightarrow M$, which is not a diffeomorphism if we consider it as a map between the smooth manifolds $(M,A) \rightarrow (M,A)$, then we can define a distinct smooth structure, say A', on $M$ by composing the coordinate charts of $M$ with $\varphi$.

Now consider $\varphi: (M,A') \rightarrow (M,A)$. Which this respect to these smooth structures, $\varphi$ will be a diffeomorphism. So while you have a distinct smooth structure, it is not really that different.

The (quite difficult) question how many smooth structures on a given topological manifold exist up to diffeomorphism. This is what Tu talks about.