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I think the biggest advantage NNs have over CA is the fact that most CA only provide localized computation. It can take a large number of fixed iterations before information propagates to the appropriate location in the 1d/2d/3d/etc. space. Contrast this with arbitrary NN topology where instant global connectivity is possible between any elements.

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CNNs are CA if you don't insert fully connected layers, actually.

You also need to make the CNN recurrent, allow it to unfold over many steps, ensure input and output grid are same size and avoid non-local stuff like global pooling, certain norms, etc.

Either way, parent comment is correct. An arbit NN is better than a CA at learning non-local rules unless the global rule can be easily described as a composition of local rules. (They still can learn any global rule though, its just harder and you run into vanishing gradient problems for very distant rules)

They are pretty cool with emergent behaviors and sometimes they generalise very well

Yep, see my comment above and especially http://arxiv.org/abs/1809.02942

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