The iPhone 4 and 4S aren’t really that far from this idea, their case consisting of two pieces of glass and a steel frame.
Some problems immediately come to mind. While the glass material used in modern smartphones is quite tough, a smartphone made entirely out of glass could easily be damaged in a fall. Furthermore, the iPhone is chock-full of sensors, some of which might have problems functioning in a (transparent) glass enclosure – though Apple’s application addresses this problem by suggesting some parts of the glass case can be made opaque.
And the upsides? Well, Apple names a few in its patent application: “Some of the reasons for using glass over other materials are that glass is strong, stiff, and radio transparent and therefore a suitable material for an enclosure of an electronic device capable of wireless communications.”
Once I had a phone.