The difference is in the amount of current which is allowed to flow in the circuit, as follows.
Imagine we generate a spark between two oppositely charged electrodes close to each other, in air. The spark provides a low-impedance path for current to flow through the air from one electrode to another. More current means more ionization, which means lower impedance, which means more current -> more ionization -> lower impedance -> more current, etc. and if we furnish the electrodes with as much current as they want, then the spark quickly degenerates into a power arc consisting of plasma carrying hundreds of amps of current and generating enough heat via ohm's law to melt steel.
On the other hand, if we externally restrict the amount of current allowed to flow between the electrodes (say for instance with a series resistor), then we can stabilize the ionization in the gap at any level we want, and obtain for example a low-current glow discharge between them that cannot degenerate into a high-current power arc.
So... a neon tube power supply has a current-limiting resistor in series with the tube so you get only a glow discharge from the tube- whereas an arc welder power supply does not have the same sort of current limiter in it, so "striking" an arc between the welding electrodes will instantly yield a super-hot plasma carrying hundreds of amps.