I'd like to elaborate on one of the points that has already been made. My point here actually does not pertain to problems from physics where we actually get an explicit solution. Instead I would like to justify why you would want to use PDE theory in a problem which you can't solve explicitly.
One of the first things that "real" PDE theory teaches us is that the statement of a PDE does not always tell us what type of solution we should expect to the equation. Specifically, there might not be a classical solution, which means a function with all the derivatives in the equation which satisfies the equation.
In many real problems, ranging from simple problems like linear transport to complex problems like Hamilton-Jacobi-Bellman equations, these often simply do not exist. This can happen artificially, such as if we use a discontinuous initial condition, or it can be thrust upon us in various ways. In either case, even when we are looking for a numerical solution, we should really have some idea of what type of thing we're looking for. It is hard to know this without having a weak formulation in front of you.
Here are some basic questions you might ask about the true solution. Is it continuous? How many derivatives does it have? A method assuming more regularity than you have will blow up.
In a problem with both time and space, can we be sure that the solution remains continuous in space if the initial condition was continuous in space? In hyperbolic problems, we often cannot, and so we have to manage shocks. A standard example is a very simple model of traffic flow: $u_t+(1-u)u_x=0$ (where $0 \leq u(0,x) \leq 1$). A numerical method which was not carefully designed to accommodate shocks will give nonsensical results, specifically non-physical shock velocities, when it encounters a shock.
This is just one of many things that one can take from PDE theory and apply when ultimately trying to solve using numerical methods.