The text mentions two definitions of the space of physical states. One is $\mathcal{H} \setminus \{0\}$ modulo $\mathbb{C}^*$, and the other is the unit sphere in $\mathcal{H}$ modulo $U(1)$. They are equivalent, there's no distinction between using one or the other.
The unit sphere $\mathcal{H}_1$ is the subset of vectors with absolute value 1; topologically, it's equivalent to $\mathcal{H} \setminus \{0\}$ quotiented by $\mathbb{R}^*$. This is because in this latter set, the equivalence class of a vector $\Psi$ is the set of all the nonzero real multiples of $\Psi$, which contains exactly one element of absolute value 1; and conversely, every element of the unit sphere is in one equivalence class, so there's a one-to-one correspondence.
Now, $\mathbb{C}^* = \mathbb{R}^* \times U(1)$, because a nonzero complex number has an amplitude and a phase. We can think of $(\mathcal{H}\setminus \{0\}) / \mathbb{C}^*$ as being done in two steps: first we quotient by $\mathbb{R}^*$ and get the unit sphere, and then we quotient by $U(1)$ and get the space of rays.
This is a bit extra, but strictly speaking $(\mathcal{H}\setminus \{0\}) / \mathbb{C}^*$ and $\mathcal{H}_1 / U(1)$ are homeomorphic, but they are not the same set. Both are sets of equivalence classes; but each equivalence class $[\Psi]$ in the first quotient has elements with arbitrary amplitude, while the equivalence class in the second quotient only has unit vectors. Still, there's a one-to-one correspondence between the two quotients, and I'm sure that there are topological theorems assuring that everything is continuous.
