Thinking about $i^\ast \mathcal O_{\mathbb{P}^n}(1)$ as $\mathcal O_C(1)$, there is a natural exact sequence
$$0 \to I_C(1) \to \mathcal O_{\mathbb{P}^n}(1) \to \mathcal O_C(1) \to 0.$$
Assuming $C$ is non-degenerate, we obtain an exact sequence in cohomology
$$0\to H^0(\mathcal O_{\mathbb{P}^n}(1))\to H^0(\mathcal O_C(1))\to H^1( I_C(1))\to 0.$$
It folllows that $h^0(\mathcal O_C(1))\geq n+1$, with equality if and only if the $H^1$ group above vanishes. The vanishing of this $H^1$ is called linear normality of the embedded curve; however, the definition of linear normality is simply that every section of $\mathcal O_C(1)$ is a restriction of a section of $\mathcal O_{\mathbb{P}^n}(1)$. We've thus really only given the obstruction to equality a name, rather than isolated a fundamental geometric property.
In practice, this sequence is still useful: often you can analyze the ideal sheaf $I_C(1)$ by using other methods, such as a resolution of the ideal of $C$.