What is the difference between a 120v 10 amp transformer and a 240v 5 amp transformer?
The two main factors influencing the voltage and current rating of a transformer are:

Voltage: This is determined by the winding insulation.  A 240V winding needs to withstand twice the voltage potential of a 120V winding.

Current: This is determined by the wire size or gauge.  A 10A winding needs to be able to handle twice the current of a 5A winding.


There are other factors such as the turns ratio, core material and saturation characteristics, etc.
If you connected a 120V rated transformer to 240V you may or may not have problems such a insulation breakdown, shorts, smoke, and burning.  It just depends on the safety margin.  But it's generally a BAD idea to exceed the stated limits.