What makes the Jones polynomial dramatic, I think, is not that it is a polynomial invariant per se, but that it came from an unexpected source where nobody had thought of looking. Indeed, there's still no conceptual mathematical explanation for why we should expect knot invariants to come from such a source.
Jones was working on the Potts model in statistical mechanics (how could this possibly be related to topology?). In this context, it was relevant to study representations of the braid group with n strands Bn into the Temperley-Leib algebra TLn. The miracle now is that the Markov trace of the representation of a braid, suitably normalized, is invariant under Markov moves, and is therefore an invariant of the knot obtained by closing the braid. Why should it be a knot invariant, conceptually? Nobody knows.
What makes it even more amazing is that the Jones polynomial turns out to fit into a family with the Alexander polynomial, which is the archetypical algebraic knot invariant, which heuristically suggests that the Jones polynomial is something important which we should be looking at, and which should probably have a sensible topological interpretation.
The Jones polynomial, I think, is "mathematics we can calculate" as opposed to "mathematics we understand", even now, 25 years after its discovery. Yet it turns out to be tremendously powerful, and to have deep connections with other parts of mathematics.