Well, the thing about electricity is- it's more than just volts and amps. Power-AKA heat and thermal considerations comes into play.
If you are conducting both voltage and high amperage, the losses heat the component. It must then transfer that heat to something else- heatsinking- in order to not overheat and thermally break down.
The barrel style epoxy covered diode uses the legs (the metal wires) to transfer heat from the diode junction- hence they are thicker wires to help transfer that heat typically to the circuit board and it's copper traces.
The stud mounted diode is meant to attach to a heatsink directly, thus handling far more power (again voltage times current).
Again, a diode has some level of voltage drop across the junction- that drop= heat. How much heat is dependent on the current and the higher the current, there is slightly more drop and more heat.
