I wonder why the Artin conjecture is so important. The only reason I could figure out is that one could use the holomorphy of Artin L-series and Weil's converse theorem to show modularity of two-dimensional Galois representations.


Are there other reasons?



Applications of Artin's holomorphy conjecture
From a modern view-point, its importance stems from the relationship with modularity/automorphy. Namely, a stronger conjecture, due to Langlands, should be true:
if $\rho:G\_K \to GL\_n({\mathbb C})$ is a continuous irreducible representation, for some number field $K$, then there is a cuspidal automorphic representation $\pi$ of $GL\_n({\mathbb A}\_n)$ such that $\rho$ and $\pi$ have the same $L$-function; in short,
$\rho$ is determined by $\pi$.


This is a non-abelian class field theory. In the case $K = {\mathbb Q}$ and $n = 2$,
it says that $2$-dimensional irred. cont. reps. of $G\_{\mathbb Q}$ are classified by either
weight one holomorphic cuspidal eigenforms (if the rep. is odd) or Maass cuspidal eigenforms with $\lambda = 1/4$ (if the rep. is even). The odd case is now a fully proved theorem
(of Deligne--Serre to go from the modular forms to the Galois reps., and of Khare, Wintenberger, and Kisin to go the other way), while the even case is still open. (If the Maass form is dihedral, one can construct a corresponding dihedral Galois rep. --- this is due to Maass himself --- but otherwise that direction is open; if the image of $\rho$ is solvable, then one can construct the corresponding Maass form --- this is due to Langlands and Tunnell.)


As you note, in the two-dimensional case over $\mathbb Q$, Weil's converse theorem shows that Langlands' conjecture is equivalent to the Artin conjecture. (In fact, proving this over any number field $K$ was one of the goals of the famous book of Jacquet--Langlands).
In general, the two are also very closely linked, so that no modern number theorist thinks about one without the other. In general, one working principle of modern number theory is that the only way to establish holomorphy of $L$-functions arising from Galois representations is by simultaneously proving automorphy.


