In Ravi Vakil's book, the Riemann-Roch theorem for an invertible sheave $\mathscr L$ on regular projective curve $C$ is stated as
$$ h^0(C, \mathscr L) - h^1(C, \mathscr L) = deg(\mathscr L) - g +1 $$
where $g$ is the genus of the curve. To me, Riemann-Roch theorem gives an upper bound on $h^0(C, \mathscr L)$ given the genus of $C$ and degree of $\mathscr L$.
Is there any other reason why this theorem is considered so important? What really makes this theorem so significant?
This is not only an lower bound : if $\deg \mathscr L \geq 2g -1$ then $h^1(C, \mathscr L) = 0$ so it is an exact computation in this case. Estimation of $h^0(C, \mathscr L)$ is very useful, see e.g the book by Miranda, "Introduction to Riemann surfaces and algebraic curve", chapter 7 where a lot of geometrical applications of Riemann-Roch are used.
Here are few immediate applications of Riemann-Roch :
1) $\mathscr L$ is very ample if $\deg \mathscr L \geq 2g+1$. Indeed, it is equivalent that $\ell(D -p - q) = \ell(D) - 2$ (where $\mathscr L = O_X(D)$ ) and follows easily from Riemann-Roch and Serre duality.
2) If $C$ is a projective curve, then for any point $p \in C$, $C \backslash \{p\}$ is affine. Indeed, take $D = (2g+1)\cdot p$, by 1) $D$ is very ample. So there is an hyperplane $H \subset \Bbb P^n$ such that $H \cap \phi_D(C) = p$ where $\phi_D$ is the map associated to $D$, so $C \backslash \{p\} \subset \Bbb P^n \backslash H \cong \Bbb C^n$ is affine.
3) If a curve has genus zero, then it is isomorphic to $\Bbb P^1$. Indeed $p$ is very ample so there is an embedding $X \to \Bbb P^1$, and a non-constant morphism between complete curves is surjective so it is an isomorphism.
4) Similarly a wise use of Riemann-Roch gives you the following : any smooth curve of genus one is an torus $E = \Bbb C/ \Lambda$, any curve of genus two is hyperelliptic, any smooth curve of genus three as a plane quartic in $\Bbb P^2$ or an hyperelliptic curve, any curve of genus $4$ is hyperelliptic or the intersection of a quadric and a cubic surface. More details are in the book by Miranda, chapter 7.
