Varicella zoster virus is the cause of chickenpox and zoster often produce a highly contagious and mild disease in children but if an adult acquires a primary VZV infection, it may lead to more severe disease.
Not sure if the answer to this is actually known? However, as far as I know (do correct me if incorrect), children are more frequently exposed to the Varicella virus (even if the initial exposures do not result in a successful infection). So their immune system is a little more primed and thus the illness is usually mild.
In adults, they are more rarely exposed to Varicella, so if there is a successful infection at a time after childhood, then this occasionally results in a more lengthy and intense symptomatic illness.
Issues of elderly and immunocomprised adults, plus reactivation of shingles etc complicate things of course.
This may be related to immune response to VZV infection and various virus-host cell interactions. There are many evidences in favor to the crucial role of cell-mediated immunity in VZV infection. It was observed that an impaired cellular immune response favors severe forms of varicella and is associated with higher rate of complication and lethality. Likewise, the spread of the VZV in the body, after secondary viremia, is mostly by the intracellular route, rather than by release of cell-free virus in vesicular fluid. It is established that cell-mediated immune responses enables virus transport via T cells to the skin but also lysis of target cells by cytotoxic T cells stimulated with VZV antigens. Natural killer cell and antibody-dependent cellular toxicity to VZV have been reported The strongest cell-mediated immune responses to VZV appear to be in early adulthood with declining with advancing age, starting at age 50. Hypothetically, it is possible, that appearance of the excessively strong and robust cell mediated immune response to VZV in adults may be sometimes harmful for the host leading to excessive damage of VZV infected cells which consequently more severe disease and higher percentage of complications.
On the other hand, it is possible that adults have an enhanced primary VZV viremia, which predisposes severe forms of disease and does not leave much time to the immune system to react. However, this requires further study for better understanding of differences in age-specific antiviral immunity in children and adults.