This incorporation [of DA into climate models] occurs at a number of stages of the model development, including parametrization of sub-grid scale effects and model tuning. The process is not, however, done systematically and current practice is not thought of as "data assimilation." There seems to be a growing realization that DA will have a significant role to play in future climate model development. This is, in part, driven by the need to quantify uncertainty in the model predictions. Nevertheless, there is not a consensus as to how DA should be used in these large-scale climate models. (source: http://www.samsi.info/working-groups/data-assimilation-ipcc-level-models-climate-uq )
Coupled models
Weather models may represent the ocean as a parameterized surface flux (of momentum, moisture, etc) or perhaps handle it through data assimilation. Climate models typically couple the atmosphere model to an ocean model and simulate the ocean as well. The climate models in actuality are typically suites of models that all communicate with each other. You may have a model for atmosphere, one for soil, one for ocean, one for vegetation, one for chemistry, etc. A weather model may have these features, but typically as parameterizations.
Spatial coverage
Weather models vary from global models to very localized regional models, which can in some cases be very idealized. Climate models tend to be global. This doesn't change the physics involved, but can influence the specific forms of the equations. A global model will solve in spherical coordinates and many use spectral methods. Regional weather models will use Cartesian coordinates and may make other assumptions that simplify the physics for the specific purpose the model (e.g. a storm scale idealized weather model may neglect Coriolis).