A 2020 arxiv posting of Nosaka (An $SL_2(\mathbb{R})$-Casson invariant and Reidemeister torsions) defines an $SL(2,\mathbb{R})$ Casson invariant. As Charlie's answer suggests, the approach is inspired by Johnson's unpublished work.
Perelman has proved Thurston's geometrization conjecture, which says that every irreducible 3-manifold decomposes along its canonical decomposition along tori into pieces, each admitting a geometric structure. A "geometric structure" is a nice riemannian metric, which is in particular complete and of finite volume.
There are eight geometric structures for 3 manifolds: three structures are the constant curvature ones (spherical, flat, hyperbolic), while the other 5 structures are some kind of mixing of low-dimensional structures (for instance, a surface $\Sigma$ of genus $2$ has a hyperbolic metric, and the three-manifold $\Sigma\times S^1$ has a mixed hyperbolic $\times S^1$ structure).
The funny thing is that geometrization conjecture was already proved by Thurston when the canonical decomposition is non-trivial, i.e. when there is at least one torus in it. In that case the manifold is a Haken manifold because it contains a surface whose fundamental group injects in the 3-manifold. Haken manifolds have been studied by Haken himself (of course) and by Waldhausen, who proved in 1968 that two Haken manifolds with isomorphic fundamental groups are in fact homeomorphic.
If the canonical decomposition of our irreducible manifold $M$ is empty, now we can state by Perelman's work that $M$ admits one of these 8 nice geometries. The manifolds belonging to 7 of these geometries are well-known and have been classified some decades ago (six of these geometries actually coincide with the well-known Seifert manifolds, classified by Seifert already in 1933). From the classification one can see that the only distinct manifolds with isomorphic fundamental groups are lens spaces (which belong to the elliptic geometry, since they have finite fundamental group).
The only un-classified geometry is the hyperbolic one. However, Mostow rigidity theorem says that two hyperbolic manifolds with isomorphic fundamental group are isometric, hence we are done. Some simple considerations also show that two manifolds belonging to distinct geometries have non-isomorphic fundamental groups.
Therefore now we know that the fundamental group is a complete invariant for irreducible 3-manifolds, except lens spaces.
Best Answer
Wikipedia's description of the Casson invariant gives the first important reason to study it. As an invariant that comes from the $\text{SU}(2)$ representation variety of $\pi_1(M)$, it reveals in particular that $\pi_1(M)$ is non-zero. At the time, before Perelman's proof of the Poincaré conjecture and geometrization, there was a lot of mystery about potential counterexamples to the Poincaré conjecture. For instance, one speculation was that the so-called $\mu$ invariant could reveal a counterexample. Since the Casson invariant lifts the $\mu$ invariant, and since it proves that $\pi_1(M)$ is non-trivial when it is non-zero, it is one way to see that the $\mu$ invariant can never certify a counterexample to the Poincaré conjecture. (Of course, no we know that there are no counterexamples.)
A second fundamental reason to study the Casson invariant is that it is the only finite-type invariant of homology spheres of degree 1. Many interesting 3-manifold invariants are finite-type, or (conjecturally) carry the same information as a sequence of finite-type invariants. This is known more rigorously at the level of knots; for instance, the derivatives of the Alexander polynomial, the Jones polynomial, and many other polynomials at $1$ are all finite-type invariants. At the level of knots, the second derivative of the Alexander polynomial, $\Delta''_K(1)$, is known to be the only non-trivial finite-type invariant of degree 2, and there is nothing in degree 1. So it means that this invariant appears over and over again as part of the information of many other invariants; there are many different definitions of the same $\Delta''_K(1)$. The same thing should happen to the Casson invariant, and indeed there are already two very different-looking types of definitions: (1) Casson's definition; (2) either the first LMO invariant or the first configuration-space integral invariant.
A third fundamental reason is that Casson invariant has an important categorification, Floer homology, which is the objects in the theory whose morphisms come from Donaldson theory. One wrinkle of this construction is that it is only a categorification of one of the definitions of Casson's invariant, Casson's definition. If Casson's invariant has many definitions, then it might (for all I know) have many different categorifications.
If your question is meant in the narrow sense of what topology you can prove with the Casson invariant, then you can definitely prove some things but only (so far) a limited amount. However, if you are interested in quantum topological invariants in their own right, and not just as a tool for pre-quantum topology problems, then the Casson invariant is important because it is a highly non-trivial invariant that you encounter early and often.