Put simply, why do some diodes such as most Zeners and Schottky diodes have a glass package as opposed to the more traditional plastic package?

Is it ease of manufacturing, thermal properties, or some other electrical phenomenon?

Early semiconductor diodes were mostly glass packaged which provided the advantage that they were hermetic and did not depend on passivation of the chip to survive heat and humidity. The glass package also allows a very high operating temperature. Early devices such as the 1N34A (germanium) and the 1N914 as well as the 1N7xx Zener series became very popular and inexpensive. 

Plastic-packaged devices were developed to reduce costs where high performance was not so important. 

For example, the glass 1N4148 has a maximum junction temperature of 200 °C compared to only 150 °C for the plastic-packaged 1N4001.

Ceramic packaged diodes have also been produced. 
