Why is it important to attempt to find an analytic solution for any theoretical model? It usually happens that many of the hamiltonians written to model the system may not usually have exact solutions. But, why are people crazy about finding them when they can always find a to numerically solve it?
Well, obviously an exact solution is better (I would consider) than a numerical one. Having an equation is something that people can work with much better than having a set of data, or a graph. Also, most numerical models, at their foundation, still rely on starting with an exact solution and expanding on them.
The hydrogen atom is a classic example. Starting from the exact form, we expand them by introducing perturbations and studying how the atom behaves under, say, an electric field. Numerically calculating the solution of the effect of the perturbation by an electromagnetic wave is much harder, since we have no foundation to start with. It also wouldn't really shed any light on why the solution is the way it is.