For a long time I've wondered why it was/is necessary to have separate solutions to the field equations for the interior and exterior metrics of a Schwarzschild black hole. Is there something weird going on at the event horizon that makes a single solution mathematically impossible? Has anyone ever found a single solution? It just seems odd that it would be necessary to find two separate solutions and then join them at the horizon. I'm not a mathematician so I would appreciate a general, non-technical answer if that's possible.
They're not two separate solutions. It's just that when you express them in a particular set of coordinates, the Schwarzschild coordinates, the coordinates misbehave at the horizon. There are other coordinates, such as the Kruskal-Szekeres coordinates, that don't have this problem.
The other thing to realize is that it just isn't normally possible to cover a manifold with one set of coordinates and have the coordinates be well behaved everywhere. If you impose x-y Cartesian coordinates on North America, they will end up misbehaving if you try to extend them to cover the whole globe. Latitude-longitude coordinates misbehave at the poles.
By the way, the two regions of spacetime that you have in mind are only half of the maximally extended Schwarzschild spacetime.