Answer:
The neutral wire is the wire that the electricity flows through once a circuit is completed, such as when you plug something in. It has no electricity flowing through it until electricity is flowing through some device and back out to your circuit breaker.
The ground wire, on the other hand, will hopefully never have electricity flowing through it. It's only there as a safety mechanism: if, for some reason, electricity escapes the bounds it is supposed to stay in when powering a device, the electricity is supposed to go through the ground wire which completes a circuit and trips the breaker before someone touches the device that has rogue electricity flowing into it.
That's why the ground wire is attached to the metal casing of devices so that if electricity is going into the case for some reason, it'll harmlessly flow through the ground wire instead of becoming a booby trap, you know, for boobies.
Other answers
A GROUNDING CONDUCTOR (also know as an EARTH WIRE in some countries) is a separate conductor which seems to be doing nothing but is in fact an important protective wire.
It is there, ready to take the current away to Ground/Earth if it, or the body/frame of the electrical appliance or device it is connected-to, makes contact with any "Hot" (= "Live" ) wire.
In USA, Canada and countries using a similar 60 Hz system, 120 Volts household electric power is carried in 2 current carrying conductors: a "Hot" (= "Live") and a "Neutral".
In Europe and countries using a similar 50 Hz system, 230 Volts household electric power is also carried in 2 current carrying conductors: a "Live " (= "Hot") and a "Neutral".
So, IF one of the "Hot" or "Live" conductors should contact the metal frame or housing - perhaps because the appliance got damaged by being dropped from a table, or similar accident - the third wire which is the "Ground" or "Earth" wire, which runs directly from the housing to the grounding [or earth bus] in the fuse or breaker panel, will in effect cause a short circuit which should blow the fuse or trip the breaker.
This third wire guarantees a current path back to the load center where the fuse or circuit breaker protecting that circuit is located, in the event the Hot wire should be in contact with the frame, but the other ["Neutral", also sometimes known as a "Return"] conductor should happen to be cut, disconnected, or open.
The idea is to guarantee that if a part of the appliance should become electrically "Hot" - which could be fatal to anyone who then came into contact with it - the "short circuit " current going into the Ground wire should cause the circuit protection device [a fuse or a breaker] to turn off the flow of current to the appliance.
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In the absence of an Earth wire (= Ground wire in US/Canadian English), if the appliance suffered some damage that caused a short circuit between the high voltage "Hot" lead and the case of the appliance, the damage would make the case live and it would cause an electrical shock to anyone who touched it.
If the case is earthed by using a Ground wire (= Earth wire in British English), if that same damage occurred the Hot lead would immediately be shorted to ground and in theory cause the fuse or circuit breaker to open, thus eliminating the danger of a live case.
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As always, if you are in doubt about what to do, the best advice anyone should give you is to call a licensed electrician to advise what work is needed.
Before you do any work yourself,
on electrical circuits, equipment or appliances,
always use a test meter to ensure the circuit is, in fact, de-energized.
IF YOU ARE NOT ALREADY SURE YOU CAN DO THIS JOB
SAFELY AND COMPETENTLY
REFER THIS WORK TO QUALIFIED PROFESSIONALS.