A small AC-DC converter like this is almost likely a flyback topology.
AC-DC converter rated for 120VAC and plugged into 240VAC will not work almost certainly. There should be an overvoltage protection circuit inside. The purpose of this circuit is not to protect the wall wart, but to prevent a fire. It's designed to trip somewhere above 120VAC, but most likely below 240VAC. The overvoltage protection circuit usually consists of a MOV and a fuse.
AC-DC converter rated for 240V and plugged into 120V might work. But there are failure mechanisms in this case too. To deliver 2.4W, the converter would pull 0.01A off 240VAC supply, or 0.02A from a 120VAC supply. Factor of 2. Granted, additional 0.01A is a very small (probably negligible) increment for a power supply. But in a larger power supply a 2x current can lead to a blown fuse, or higher copper loss in the transformer (and burn it), or higher loss in the switching transistor (and burn it).