AC: Why differentiate between Ground and Neutral?
If wires were 100% reliable and had zero resistance, there would be no difference between the neutral (groundED conductor) and the safety ground (groundING conductor). Neither condition applies, however.
Even if the neutral-grounded and safety-grounding conductors are connected at the breaker panel, a current-drawing appliance some distance from the box may cause significant voltage drop in the neutral-grounded conductor. Having any exposed parts of the device connect to ground using a separate safety-grounding conductor will avoid the voltage on its end of the neutral wire from appearing on its exposed parts.
Additionally, using separate conductors ensures that a variety of single failures may occur without creating an immediately dangerous situation (though a second failure which occurs without the first having been corrected could be immediately dangerous).
If exposed parts of a device are connected to the safety-grounding conductor, and a hot wire within the device accidentally touches those parts, short-circuit currents should trip the breaker.
If the hot wire fails between the breaker panel and device, the device would get no power, but there would be no dangerous voltages anywhere near the device.
If the neutral-grounded wire fails, the neutral wire in the device may be only a few ohms separated from direct hot potential, but no current would flow through it, and no path would exist from it to anything the operator might touch. Exposed parts would still be safely connected to the safety-grounding conductor.
If the safety-grounding wire fails, the device would no longer be protected against the possibility of a hot wire touching the case, but no immediate danger would be created.
If the case were not connected to anything, failure #1 would create an immediate potentially-lethal situation; if it were connected to neutral, failure #3 would create an immediate potentially-lethal situation. With both wires present, however, a single failure will not create immediate danger.
TL;DR:
The ground wire is a safety feature to keep you safe in case things aren't working right.
You have a neutral wire as a current conducting wire to provide power.
You have the ground wire as a safe ground point for equipment with conductive (metal) housings and as a safe short circuit path for current when things go bad.
Now, some background. In the US, power is delivered to the house at higher voltage and is stepped down to provide 230 VAC with a center tap.
The neutral is connected to the center tap.
From the two ends of transformer output you have 230VAC.
From either end to the center tap you have 115VAC.
There are thus 2 circuits that provide 115VAC. These 2 circuits each provide power to half the lights and half the outlets in the house.
The neutral is thus floating, and at some unknown voltage above the voltage of the (literal) ground beneath your feet. Touching the neutral would be very dangerous. Touching either of the live wires is also very dangerous.
To keep the neutral from floating, it is connected to the house's ground - there's a large metal conductor in the ground beneath the house that provides a real connection to the real ground.
There are two points of danger when dealing with a power system.
One is the danger of connecting yourself between two voltage carrying lines - this will obviously cause current to flow through your body.
The other danger if of connecting yourself between a voltage carrying line and the ground - literally, the ground beneath your feet. If the power system is not grounded it will always have a voltage difference as measured to the ground.
The first danger can be worked around by never touching more than one wire at a time - usually pretty easy to do.
The second is much more difficult. If you touch any wire from an ungrounded power system, there will be a voltage difference between it and the ground and current will flow through your body=ouch/dead.
To reduce this second danger, power systems are grounded.
In the US, you ground the neutral wire. It is now (nearly) at ground potential. Now, there's one wire that it should be safe to (accidentally) touch. This is the reason to connect the neutral to ground.
The two live wires are now at 115VAC as measured to ground, but there's only one live wire in each outlet, so the wiring is somewhat safer - there's only one wire in the outlet box that can kill you.
BUT we aren't through yet. If there is a large current flowing through the neutral then (thanks to Ohm's law) there will be voltage difference between it and ground, so neutral is no longer really at ground potential.
Given that the two 115VAC circuits in an American house can never be balanced, there is almost always a current flow through the neutral line therefore it is not really at ground potential.
Now, imagine you are using a device with a grounded metal housing. If you are using the neutral as a safety ground, then the housing isn't really at ground potential so you get a (hopefully only) low level tingle if you touch the housing - not good, can still hurt.
If there's a short from the live wire to the metal housing then the voltage on the housing will rise== Ouch,Ouch, Ouch. If the neutral wire nows breaks in the power cord or has a bad connection in the outlet then metal housing is now at line voltage= dead user.
Now, imagine the same device with a safety ground wire. The safety ground is connected to the metal housing. Since there's never any current flowing through the safety ground (except when it is protecting you from a short circuit) the housing of the device is really at ground level=perfectly safe, no tingle.
If there's now a short from the live wire to the housing, the voltage on the housing will only go up a little bit (resistance of the ground wire) before the circuit breaker disconnects. The voltage might get high enough to tingle, but not enough to kill= user gets to keep on living.
Based on the various premises, this is most likely a question about the US NEC (National Electrical Code) requirements, I think.
Why differentiate between Wire-2 and Wire-3 at the power socket but then connect them further down the line?
Because if you connect them further upstream from the main panel, then you have a normal return current path through the grounding wire, which creates an unsafe situation for anyone touching it or anything connected to it, which is a lot of metal casings. As further detailed in one decent book on the topic.
The neutral is a grounded conductor by virtue of the connection at the service, but is not a grounding conductor because it is not used to connect anything else to ground. It is only used to carry the normal load current of lights, outlets, or other devices that are connected from phase to neutral. The grounded conductor remains isolated from ground everywhere except for the bond at the service. If more than one connection to ground is made, load neutral currents will divide between the grounded conductor and the EGCs (equipment grounding conductors). This can result in continuous current flow on conduit systems or metal structures and piping, which can cause electrolytic corrosion and interference with sensitive electronic equipment due to radiated magnetic fields.
Actually the US setup isn't all that foolproof as a pig pole (transformer) is shared by several houses (in the burbs) and an open neutral in one house creates the following current return path through the grounding of a nearby house, something that's not terribly easy to debug (image from the same book):
As for the other question:
Why is it safe to touch Wire-3 but not Wire-2, or why can Wire-3 provide a level of protection that Wire-2 can't?
Well, it's safer most of the time. It's surely just as unsafe during a fault. The same book says (p. 104):
Never assume that a grounding electrode conductor is dead.
Finally, this NEC-mandated setup is called a TN-C-S earthing system in IEC lingo. In Europe (including UK), particularly in urban areas, the TN-S system, in which the earth is split from the neutral all way to the substation is common.