3-body problem is unsolved math problem. There is equation, but no one knows how to solve it, and the only way to deal with it is number crunching. Stable orbits are solutions to the problem, but partial ones. The more solutions are known there more possibility to find more generalized solution. Maybe even the general solution for the problem.

I do not know how it can change astrophysics (its just newtonian gravitation, not einshteinian), but it can bring new methods/ideas to mathematics, then improved math will change everything. Maybe.

The problem isn't unsolved. It just doesn't have analytical solutions. But the mere fact we can simulate trajectories means we can solve it.

From engineering point of view, maybe the problem is solved. From maths point of view it isn't. For example, one cannot say for sure, are these trajectories periodic or no. 100-digit precision says "yes", but there is no guarantee, that 101-digit precision wouldn't say "no", they are not periodic.

what kinds of problems can we solve with an analytic solution to the 3 body problem instead of a numerical one?

Given this other reply: https://news.ycombinator.com/item?id=14280369

We'll probably be able to use these techniques on other problems in the future as well.