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Let’s deal with the computational intensity, first.State-of-the-art weather and climate computer programs differ only in the time step of the simulation and in the granularity of the Earth surface 3D grid [wiki-nwp].We know and use the statistics but cannot model the process each carbon atom undergoes.
The simple explanation is that weather is chaotic noise and climate is the statistical characteristics of the noise.
The important difference is that we really are interested in the noisy weather five days from now when we have plans whereas in most engineering problems we care about the signal and to find that we only need to know the statistics of the noise.
Another good example is carbon 14 dating [wiki-rcdating] used in archaeology, history and other fields.
Weather models assume a horizontal grid size of a few kilometers (or miles) covering the Earth surface while climate models have a horizonal grid size of a few hundred kilometers. Figure 3 shows some of the physical phenomena which are calculated at each grid point.
Weather simulations recompute parameters every few minutes while climate models compute them every tens of minutes. Snyder, Evolution of global temperature over the past two million years, Nature, 2016 [wiki-nwp] https://en.wikipedia.org/wiki/Numerical_weather_prediction.