How much space does a 220V AC input power supply save compared to a 120V one?
The weight is due to the transformer. Higher is the frequency, smaller the transformer is. I have a hobby MMA welder, it is a forward converter with two IGBTs and two diodes, it uses SG3525 PWM controller with a PWM frequency near 50kHZ. It is rated at 140A for 30% intermittent duty.

A smaller welder, TIG/MMA Fronius transpocket TC1500 has a full bridge converter with 4 FET transitors, fast gate drive circuitry clocking at approx 140kHz , a tinny transformer, MCU/FPGA control. It can weld almost continuously at 140A, fully protected. 

The price between these two welders is 1:5. Higher power densities are possible with higher price (TC1500 is almost 15yrs old, mine hobby welder is 3yrs old) and even better efficency - the TC1500 doesn't heat as much.

So, conclusion: higher switching frequency -> less weight -> higher price.