Why should I care about the Jones polynomial?
Your question presupposes that people were excited about the Jones polynomial because it would help them to classify/distinguish knots. In fact, I suspect the interest came from the fact that this knot invariant was originally defined using operator algebras (rather than in the more combinatorial way people usually define it now). Operator algebras had grown out of an attempt to formalise quantum mechanics/QFT, and it was surprising that a knot invariant should appear naturally in a completely different subject. At around the same time, other manifold invariants (Donaldson invariants, instanton Floer homology) appeared that were also inspired by constructions in physics. These other manifold invariants had very definite topological consequences, solving huge open questions in 4-dimensional topology.
Witten showed that all of these invariants (including the Jones polynomial) can be obtained formally by performing path integrals. For example, roughly speaking*, the Jones polynomial can be obtained by looking at all possible connections A on a suitable bundle on your 3-manifold, taking the trace of the holonomy of A around your knot, multiplying by e^(iCS(A)) where CS is the Chern-Simons invariant of A, and then integrating the result (over the infinite-dimensional space of all connections, which is a kind of path integral). Whether or not you care about physics, that is a pretty cool way to define an invariant.**
This led mathematicians to study topological quantum field theories, and the associated manifold invariants. In particular, Khovanov was led to discover his homological refinement of the Jones polynomial, which is extraordinarily useful as a knot invariant: it is functorial under cobordisms of knots, so can be used to study surfaces in 4-manifolds, for example giving lower bounds on slice genus of knots.
So if you're looking for topological results in pure knot theory that wouldn't exist if it weren't for the Jones polynomial, you should check out Khovanov homology and its applications.
(*) I corrected what I had written earlier (it's been a long time since I looked at Witten's paper).
(**) The reason I think this is a cool way to define an invariant is that, since the thing you're integrating is covariant under diffeomorphisms, provided your path integral measure is covariant under diffeomorphisms, it would be obvious that you get an invariant. Of course, even if you knew how to define these measures, the devil is in proving that they are invariant (often, classical symmetries of a Lagrangian do not descend to a symmetry of the quantum theory, and the theory is called anomalous).
As a historical note (others may have had a different perspective - I was a graduate student when the Jones polynomial made its appearance), when it came out there was some mild excitement because the Jones polynomial could distinguish some pairs of knots which the previous invariants (Alexander polynomial) could not. There was some excitement that the invariant came from operator algebras by way of braid theory, and much more excitement because there seemed to be a connection to physics (by way of a "diagram calculus", which, in truth, was nothing new in knot theory, and came via Conway polynomial). This last excitement was more social since there was a perception that the physicists were the cool kids, and so us peasants would become cooler by doing things related to physics in some (albeit tenuous) way.
The wet blankets responded to all this with a resounding "who cares"? The results in "classical" geometry and topology (alluded to by the OP) affected by the new wave were few and far between and did not seem to merit all the hoopla.
The most measured attitude was that of Bill Thurston, who did not get involved in the rah-rah activity, but wisely thought that everything new is worth studying, especially as the invariant did have some intrinsic value. There was some hope at the time that the Jones polynomial (by the way, at the talk in Durham where Vaughan Jones described his oeuvre to an audience of geometers and topologists, he talked about TWO polynomials, one named V and one named J) would be more powerful than it would be. As pointed out by Jonny Evans, the appearance of the Jones polynomial spawned an industry of quantum invariants, to which much of the above applied (with the extra complexity that, unlike the Jones polynomial, which is easy to define, and easy to compute for small knots (otherwise the problem is $\#P$ complete), many of the newer invariants seemed completely intractable, and a lot of the theory was showing that some uncomputable invariant was equivalent to some other uncomputable invariant.
Since then, some order has entered the chaos, as, again, pointed out by Jonny.
There have been some topological applications of the Jones polynomial and its various generalizations. I believe that these applications increased the interest in these invariants by topologists.
One application was to the Tait conjectures. Jones used his polynomial to give lower bounds on the bridge number of links; see Proposition 15.6 of
Jones, V. F. R., Hecke algebra representations of braid groups and link polynomials, Ann. Math. (2) 126, 335-388 (1987). ZBL0631.57005.
See also the Morton-Franks inequality Theorem 15.1 in the paper estimating the braid index in terms of the HOMFLY polynomial.
Another source of interest is the volume conjecture which has received much attention due to the possible connection with geometric invariants of knots. There seem to be many other connections of the colored Jones polynomials with other more geometric invariants (cf. the slope conjecture and noncommutative A-polynomial).
As mentioned in the other answers, the biggest development was the definition of Khovanov homology, a categorification of the Jones polynomial, which led to a new proof of the Milnor conjecture. Khovanov homology is known to distinguish the unknot, and can be used to give invariants of transverse knots.