The difference between 0v and Neutral in an electrical/electronic circuit?

Neutral is the return path for AC current. Usually it is physically connected to earth / ground at the electricity substation, so in principle it's at the same voltage as ground, but in practice it won't be due to the current going through it and electrical noise on it.

In a DC power supply there is no defined relationship between 0V, the return point for DC current (often called ground and often connected to ground), and mains neutral. 0V may be completely isolated from neutral or it may be at a very similar voltage (especially if 0V is connected to ground). But - you would *never* connect 0V to neutral in a piece of equipment. The reason is that electricians can make mistakes. Suppose someone wires a mains outlet wrong and swaps the live and neutral wires? When you plug in your equipment 0V is now connected to live. Not a good situation!

Edit:
Adaviel summed it up well. The supply input and the low voltage section are different circuits; different circuits have different reference points.

By a "DC transformer" I guess you mean a DC power supply like for a laptop or modem.

There is almost always isolation between the input and output, so that the output pins have no connection to the input pins. They are not part of the same circuit. To get a sensible voltage reading, you must stay within one circuit.

Sometimes, the 0V output may be connected to the chassis, e.g. in consumer audio equipment, and also connected to the mains ground e.g. in PCs. That can cause trouble with ground loops (hum) if it is connected more than once

Chances are you mean a packaged transformer with a rectifier/smoothing circuit on the output to produce DC. The windings of most transformers are electrically isolated from each other (auto formers don't but they're a little unusual ) There's no electrical connection between the two windings, so it's equivalent to measuring the voltage when one of the probes isn't connecting

Side issue. There certainly were devices called DC transformers claimed to be capable to have input of DC, and output of stepped up DC without using any switching[like cap switching, or switch modes] Browsing an old catalogue[from a reputable supplier] from the 90s I once come across so called DC transformers claimed capable of stepping up the voltage of DC directly without being switch mode converters, nor rely on changing magnetic fields. claimed to use some effect produced by stacking wedges of semiconductor together to produce a voltage step up. Haven't much clue about how they worked, my best guess is a something like thermopile +heater? They were pricey, bulky for their output and with rubbish efficiencies. Never heard of them since so presume they were a dead end technology/didn't take off, or they've got called something different.

0v or zero volts is just a statement of the voltage between two points, zero. You seem to mean something more specific, but what is that?

I think you mean earth ground, I'll assume that.

Ground is a safety wire and should never have any current in it. Neutral is the return wire, and does have current flowing in it. Neutral is connected to ground at some point, but the current flowing in it means there is some small voltage drop between neutral and ground, usually less than 2 volts, 5 volt at the most.

The above is all for AC power distribution in homes and factories.

A DC supply is a different thing. It has an isolated output, + and –, and one of these is tied to ground usually, although it could be floating. If the – terminal is tied to ground, then the plus lead will be reading a + voltage with respect to ground. There is NO neutral in DC, there is a hot and a common or return. The return serves the same function as neutral in AC.

No such thing as a "DC transformer" so that part is meaningless.