It is known that Cousin problems are especially relevant in determining if a meromorphic function can be expressed as a quotient of two holomorphic functions. Indeed:
Theorem 1 Let $X$ a complex, connected variety, such that the second Cousin problem is satisfied. Then every meromorphic function is the quotient of two holomorphic functions on $X$, such that the germs of $f$ and $g$ do not have any common divisor.
In literature, this result is usually reported after
Theorem 2 Let $\phi$ a meromorphic function on a complex, connected variety $X$. Then, there exists a line bundle $L \to X$ and two sections $s_0, s_1$ of $L$ such that $\phi = s_1/s_0$.
I guess my question is: how does Theorem 1 adds any new information to Theorem 2? Indeed, every section of a line bundle is a holomorphic function on $X$ already: why would it be of any benefit for a meromorphic function to be expressed as a quotient of holomorphic functions which are not sections of a bundle in general?