You should get an identical voltage when checking the two hots against the ground as you do the neutral. If you don't, you have a problem.
As far as the two hot sides being different, it could be harmless. The normal range is 114 to 127 in the US at the service entrance. So it's within spec. A difference between the sides could just be load on the two halves of the utility transformer being different, especially if the transformer is feeding multiple houses or apartments. Or, it could be a poor connection in the neutral or one of the hot wires from the panel to the transformer. One way to maybe check that is to turn off all the load, then check the voltage difference as you add load. If it doesn't get any worse than you have, it's most likely fine, in my "guy on the Internet" estimation. If it swings up to a voltage on one side > about 130-135, then I think you have a problem with your neutral. If, on the other hand, when you turn off all the load both sides go up to 125-127, and one side drops with load, you may have a poor connection on that hot side, from panel to transformer. Of those two options, a problem with the neutral is more dangerous. Not because of shock danger, but because if your neutral opens you can see up to 240v on a 120v circuit, and that can cause many 120v to fail.
Short version? As long as neither side goes below 114 or above 127, it's probably fine. If either side swings below 110, except in a brownout, or above 130, you may have a problem.
As far as the welder goes, it's only looking at 240v across the hot leads, it'll be fine.