Why is the Hamilton-Jacobi equation important? Someone may say it is related to the Schrodinger equation. Okay, let us forget about quantum mechanics. So, if we confine ourself to classical mechanics, why is the Hamilton-Jacobi equation important or useful? We already have the Lagrange equations and the Hamilton equations for following the motion of a system. Why do we need a third equation?
 A: The Hamilton-Jacobi equation is a partial non-linear differential equation. A complete integral depends on $2n+1$ arbitrary integration constants. The complete integral defines an integral surface on which there are characteristics that are solutions to a set of first order coupled ODEs. In this way we have related a 1st order, non-linear PDE to a set of first order, coupled ODEs. 
The second importance is through the Poincar\'e-Cartan integral invariant $\theta$ we can derive pretty much all of canonical mechanics (this is an extensive subject and the standard literature is V. I. Arnold Mathematical Methods in Classical Mechanics). Just to relate to the above there is an existence theorem that proves that the Liouville 1-form in extended phase space, by the Poincar\'e lemma will admit the solution curves. 
\begin{equation}
\omega =d\theta_H,\ \ \ \ \ \ \ \ d\omega =d(d\theta _H) =0
\end{equation}
(Entering local coordinates will yield Hamilton's equations, but this is just book work).
Why do we need this as well as Lagrangian and Hamiltonian mechanics? Firstly it generalises both of these and along with the Liouville equation of motion provides a field theoretic model of classical mechanics, treating the fields as $\mathcal S$ and $\rho$. 
It also admits a natural wave interpretation through geometric optics and is the classical limit of the De-Broglie Bohm formalism. 
Lastly and perhaps the most importantly we can formulate a geometrical picture of classical mechanics on a manifold using the Hamilton-Jacobi equation. Again this topic is extensive. 
A: The Hamilton-Jacobi equation is particularly good for describing a family of solutions with different unknown initial conditions with the unknowns being parameterized in a particularly nice way (relating to conserved quantities in a nice way)
This is exactly why it is so much easier to relate to quantum mechanics, but it can be useful in other situations as well.
