TITLE: Why should I care about the Jones polynomial?
QUESTION [34 upvotes]: The invention of the Jones polynomial led to  hundreds of papers and a Fields medal.  However, as far as I can tell it had few consequences in topology.  After all, after Thurston’s work we already had algorithms to completely classify knots, so by itself a new invariant seems to be of limited value.  Given all the excitement, however, I suspect that I might be missing something.  Why was this so important?

REPLY [21 votes]: 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.