This Wikipedia page covers it well - but in short:
Neutral is the return current conductor for AC power circuits. Ground (earth) is a safety connection provided as a separate return conductor for fault currents - i.e. it does not carry current during the normal operation of any connected equipment.
Neutral and earth are connected together at the point of power generation and at the output of each transformer in a transmission system. However, since current can flow in the neutral line during normal operation, voltage can develop between it and earth at the point of consumption. As a result, in your home (for example) neutral and earth will be similar in voltage, but not quite the same.
There are a few different connection methods for ensuring the earth voltage stays close to the actual potential of the surrounding ground, with three most common in the United Kingdom.
T-T (usually older systems, also used for temporary power provision) - a stake in the ground at the transformer connects neutral and earth, and a similar stake at the consumer premises provides a local earth connection - unconnected to the transformer.
TN-S (older systems - but more recent than T-T) - transformer connected as T-T, but a separate earth cable runs from the neutral-earth bond at the transformer supply all the way to the consumer premises, providing a dedicated separate path for fault currents.
TN-CS (newer installs - last 20 years) - the neutral conductor from the transformer acts as the return path for earth currents as well. Earth within the premises is provided as a connection to neutral at the point of entry to the premises - this connection is made at one point only. Further distributed earth connections are generally made between the ground and the neutral conductor along the length of the cable in order to minimise the voltage between the ground itself and neutral.
This video covers all three systems in detail, including pros and cons of each approach.
"Carry voltage" technically is a very loose term.
Voltage is relative, meaning it must be measured relative to another point, such as ground or another conductor.
All conductors have resistance and obeys ohms law. When current flows there will be a voltage drop throughout the length of the conductor. This happens in all conductors disregarding the systems involved.
In well balanced 3 phase system the neutral current will be very minimum compared to the phase currents. That explains why voltage drop is almost zero in that wire.
Imagine a thin straight pole is poked into earth perpendicular to ground and tip of the pole is attached to 3 ropes and pulled from three equally spaced (120 degree) directions. The pole tip will stay stationary even when pulled with great force. That's what happens to the neutral in well balanced 3 phase system.
But when a rope a slightly slag or exert more force than the other two, the pole tip will move. Thus, imbalance 3 phase system will have neutral current and the proportional voltage drop along the wire.
In the real world, there is no perfectly balanced 3 phase system. That is why most of the time when neutral voltage is measured against the local earth, there will be a voltage, mostly around 6 volts for the 415 volt 3 phase systems.
Best Answer
First off you mention 240V so I'm guessing you're in the UK, it's notoriously difficult to get meter probes into a UK BS1363 socket and have them connect. The best way of connecting is to (with all due care an attention) use a BS1353 to IEC320 C13 lead (commonly referred to as a kettle lead) the female C13 connector is very easy to fit meter probes in, but again be careful. [you can get BS1363 plugs with 4mm meter sockets on the back if you want to do this safely].
Worryingly your results are self consistent and suggest a fault with your electricity supply.If you have a TT connection from your supplier (eg the power comes in on two wires overhead from a pole), then it's possible that something near either your house or the transformer is pushing the earth potential up that high. You say you've checked your earth connexion - do you mean you've checked that the sockets have a low resistance to a measuring stake you've driven in near the ground (or if you're boding it the lead/copper pipe bringing water into your house).
If you want to check your observations, you can buy cheap devices with leds/neons arranged between all the conductors. (eg https://www.screwfix.com/p/lap-ms6860d-socket-tester/91596 others are available and are better/cheaper)
or any device used to do a Part-P test would be able to check your measurements.
If your measurements are correct you'll be putting extra strain on the insulation in any of your devices that use switch mode power supplies, and I'd expect any surge suppressors in your house to have gone bang.
Either way this definitely warrants further investigation.