What are the advantages/disadvantages of climbing helmet design (foam or hardshell)?
PS: I do not intend to compare them by how much they cost.
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The British Mountaineering Council (BMC) has a downloadable helmet guidebook which addresses this question pretty well. As a brief summary:
In my opinion you should also take the hybrid helmets into account.
As far as I know, hardshell helmets are more robust than foam ones but they are heavier. That's also the reason they invented the hybrid, to get a compromise of both. Hybrid has a thinner hard outer shell (lighter than the extremely robust hardshell helmets) to protect against deformation. Nonetheless, the real safety factor comes only in combination with the foam (for hybrids).
The foam helmets (which are no hybrids) aren't that robust, you shouldn't take it as a seat for example. They are more like bicycle helmets, so they aren't that likely to be very durable for multi impacts (see this answer).
Another point is that the foam helmets are typically warmer because of the higher isolation. That might be of interest on winter tours.