Why does plastic never dry properly in a dishwasher?
In addition to the lower heat capacity (see other answer), a main reason, quite counterintuitive, why plastics don't dry well is that they're hydrophobic. That's right: they keep water sticking to them because they're water repellent (but not completely water repellent).

The reason for this strange behaviour is that any small amount of water on the surface of plastics immediately contracts to a compact droplet. This minimises contact with the plastic, but also with air, which is the problem: for efficient evaporation, you need a large air-to-liquid surface. You do get such a large surface on glass, ceramic and metal, because these are (at least when freshly cleaned) hydrophilic, so the water stretches out to a thin film coating a whole lot of the surface.

Not so with plastics. Only with some shaking will the droplets start running down the surface, and may combine with other drops and then drip off. With super-hydrophobic materials you'd be ok again because even tiny drops would immediately pour down, but most plastics are exactly at the sour spot: too hydrophobic for evaporation, but still not hydrophobic enough for a lotus effect.

Melamine resins tend to be among the better plastics in this regard, as they're still pretty hydrophilic. Still, they tend to dry only incompletely, probably because of the low heat capacity.