Twisting sheaf of projective space

Question

Let $A$ be a ring, $S=A[x_{0},...,x_{n}]$, and $X=$Proj $(S)=\mathbb{P}_{A}^{n}.$ Hartshorne defines the twisting sheaf $\mathcal{O}_{X}(n)=S(n)^{\thicksim}$. Since $\mathcal{O}_{X}(n)|_{D+(x_{i})}$ is isomorphic to $S(n)_{(f)}^{\thicksim}$ on Spec $S_{(f)}$ and for any $n$, $S(n)_{f}$ is isomorphic to $S_{(f)}$ as $S_{(f)}-$module, it look likes $\mathcal{O}_{X}(n)$ is isomorphic to $\mathcal{O}_{X}(m)$ as sheaf of modules. However, I think they are definitely not isomorphic. Where I make a mistake? And how to proof if $n\not =m$, $\mathcal{O}_{X}(n)\not \backsimeq \mathcal{O}_{X}(m)$. In general, how to know two locally free sheaf of the same rank are not isomorphic?