What are the safety and device risks associated with not connecting the grounding prong on a 120V AC device?
The primary purpose of the green-wire ground is USER SAFETY. It provides a SAFE conductive surface for the user to touch if something goes wrong inside the device and allows the live/hot node to become connected to the outside surface.

And then it provides a FAULT PATH back to ground so that the fault current will cause the fuse or circuit-breaker or Residual Current Device or Ground-Fault Interrupter to disable the live power connection.

There should be NO difference in the operation of the device, however. No device should be depending on the green-wire safety ground for ANY operational function.  All mains power should be used between the hot and neutral nodes. The Ground node is there only as a safety measure, not for any operational purpose.

Note that there are "double-insulated" gadgets which come with only 2-prong mains power plugs. The only metal parts exposed to the user have been DOUBLE INSULATED from any potential contact with the "hot" mains voltage.

Even without the green-wire safety ground connection, a modern Residual Current Device or Ground-Fault Interrupter will sense that current is going somewhere else (besides returning through the Neutral wire), and it will disable the Live power. However most circuits are NOT protected by RCD/GFI, so it is NOT safe to assume that you can do without any kind of provided green-wire ground connection.