Is there a difference between $\mathscr{O}_{\mathbb{P}^1} \oplus \mathscr{O}_{\mathbb{P}^1}(n)$ and $\mathbb{P}\left(\mathscr{O}_{\mathbb{P}^1} \oplus \mathscr{O}_{\mathbb{P}^1}(n)\right)$?
I understand that the Hirzebruch surface $\mathbb{F}_n$ is defined as the (projectivization of the) bundle $\mathscr{O}\oplus \mathscr{O}(n)$ over $\mathbb{P}^1$. So what does the notation involving putting an overall $\mathbb{P}$ around this indicate?