Distinguishing between distinct unitary irreducible representations is important from the point of view of distinguishing between different sorts of particles; the eigenvalues of the Casimir operators provide a way of doing this.
I have never properly understood why this is the case. Is there a good way of seeing why any two distinct irreps must, necessarily, have different eigenvalues for at least one Casimir?
The fact that a Casimir operator/invariant takes a single eigenvalue on an entire (finite-dimensional) irrep (over $\mathbb{C}$) follows from Schur's Lemma.
It is also clear that a Casimir takes the same eigenvalue on equivalent (=isomorphic) irreps.
So OP's question essentially boils down to whether there are enough Casimirs to distinguish different irreps (modulo equivalence). That is the topic of this related Phys.SE post.