In simple electricity terms it's like the flow of water where amps is the water flow and volts is the pressure of the water.
In welding terms amps is the "heat". To get more weld penetration or to weld thicker diameter material, you turn up the heat/amperage.
However, to increase the amps you need to have enough volts. In physics terms, amps and volts combine to determine the overall heat input. (volts x amps x 60)/travel speed = heat input in joules.
In a CV (Constant Voltage) welding power source like that used in MIG the volts can make a difference. For instance, there is a minimum voltage below which it isn't possible to do spray arc welding and only short circuit welding is possible.