What we call “differential equations” are usually not just differential equations. They also have associated initial conditions or boundary conditions.
With ordinary differential equations (ODEs), the initial conditions are often an afterthought. First you find a full set of solutions, then you plug in initial conditions to get a specific solution.
Partial differential equations (PDEs) have boundary conditions (and maybe initial conditions too). Since people typically learn ODEs first, they come to PDEs expecting boundary values to play a role analogous to ODEs. In a very limited sense they do, but in general boundary values are quite different.
The hard part about PDEs is not the PDEs themselves; the hard part is the boundary conditions. Finding solutions to differential equations in the interior of a domain is easy compared to making the equations have the specified behavior on the boundary.
No model can take everything into account. You have to draw some box around that part of the world that you’re going to model and specify what happens when your imaginary box meets the rest of the universe. That’s the hard part.