^^^ That's exactly the first thing that came to my mind teacozy!
I think there are multiple things going on so it's not just a simple answer. With chicken pox, for example, many of the complications that adults have (higher fever, more severe rash, encephalitis) can be attributed to a "too strong" inflammatory response. On the other hand, some of the complications (pneumonia and secondary skin infections) are associated with lifestyle habits such as smoking or occur more often in immunocompromised adults. So there's not just one answer.
An interesting point, while I was looking into chicken pox, was that while it is a childhood disease in developed countries; in developing countries, it often occurs later in adolescence and adulthood. Perhaps due to high infection rates with respiratory herpes - competition between viruses, maybe? There's some thought that it might be climate-related as well.