It's size one has a larger conductor than the other so that it can handle a higher amperage.
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2/0 is about 27% larger in cross-sectional area than 1/0. It therefore will be heavier, slightly stiffer, and able to carry the same amount of welding current further (or carry a higher current the same distance) compared to the 1/0 cable. Select the cable you need, based on the welding being done. You can use the heavier cable to get to the general location and then have your stinger lead be smaller for less fatigue and greater mobility.
2/O or double Ought cable will carry 400 amps for 50 feet. 1/O or single Ought will carry 350 amps for 50 feet. If you are tig welding it is doubtfull that you will ever atain this in a small shop. A unit to weld 350 or 400 amps is a large expensive unit.
The single Ought will carry 250 amps for 200 feet.