Aside from incompleteness in the equations due to, say, missing or misunderstood physics, or errors in a computer implementation of the models, a key problem is the specification of initial conditions, model parameters, and boundary conditions for the algorithm. One can prescribe the initial conditions analytically, and use boundary conditions that resemble a realistic situation.
Data assimilation takes a different tack.
One tries to obtain a best estimate of the true state of a system given a model and a finite number of observations. Typically, these observations are distributed inhomogeneously in both space and time. One may be searching simply for gridded accurate initial conditions, or one may be searching for an ongoing assimilated dataset. Atmospheric data assimilation matured significantly of the last 5 decades. This work produces datasets that are used by the climate research community.