The causal structure is basically a chaotic system, which means that NewtonIan style differential equations aren't much use, and big computerized models are. Ordinary weather forecasting uses big models, and I don't see why climate change, which is essentially very long term forecasting would different.

Climatological models and meteorological models are very different. If they weren't, then "we can't predict whether it will rain or not ten days from now" (which is mostly true) would be a slam-dunk argument against our ability to predict temperatures ten years from now. One underlying technical issue is that floating point arithmetic is only so precise, and this gives you an upper bound on the amount of precision you can expect from your simulation given the number of steps you run the model for. Thus climatological models have larger cells, larger step times, and so on, so that you can run the model for 50 model-years and still think the result that comes out might be reasonable.

(I also don't think it's right to say that Newtonian-style diffeqs aren't much use; the underlying update rules for the cells are diffeqs like that.)