If the plugs differ and both are the straight blade type, it's because something relevant to safety differs, probably which wires are present (white neutral or

If the plugs differ and both are the straight blade type, it's because something relevant to safety differs, probably which wires are present (white neutral or green safety ground) and/or the current rating. You can get in trouble if the welder expects a safety ground or neutral intended to carry current that you don't have (neither is safe substitute for the other). You can get in trouble if you connect the welder to a circuit that can deliver more current than it's designed for and something fails (it was designed to fail safely assuming that the circuit is limited to X amps by a fuse or breaker). If the dryer outlet has all the wires the welder wants (and some extra) and the same current rating, an adapter that connects the desired wires and ignore the rest should be safe as long as it's made from big enough wire.