What purpose does a 3+1 outlet/circuit provide when the neutral bus is grounded?
Yeah, the root problem is, you see they are both tied together in the panel, but you don't know what that thing is... or you don't even realize that is a thing.  

Its name is the Neutral-Ground Equipotential Bond.  Since we don't know what it is, let's get rid of it.  SNIP! 

Life without an Equipotential Bond

Now, your 2 hots are 240V apart, and neutral is in the middle, giving you 120V from neutral to either hot.  None of these have any relationship to ground at all. We cut that, remember?   So our old assumption that "Neutral is always near ground" is now void.   


Neutral is probably near ground. 
But it could be 120V from ground (I actually had that happen once, when the "snip" happened for real in a panel and I didn't know it). 
There could be leakage in the 9600V supply transformer, and L1, N and L2 could be 9600, 9720 and 9840V from ground, respectively.  


See, having the hots and neutral "rattling around out there" at random voltages isn't good.  We want to do something to peg neutral at a relationship to earth.  Hey, it could be through a 1 volt 2000A transformer - now neutral is pegged 1 volt AC from ground.  But it's pegged.  It could be a car battery - so now neutral has a 0 volt AC and 12 volt DC bias from ground. (both of which would be cool for diagnostic purposes).  But the point is, it's no longer floating.  We've pegged it.  

2kA transformers and car batteries are neat in theory, but if you have 100 houses to wire this week, you want a simple/cheap equipotential bond - one with 0V bias -a piece of copper.  

Actually, you can go cheaper still, simply by using the neutral bar as the equipotential bond, i.e. just putting neutrals at one end of the neutral bar, and grounds at the other end.  Actually, we could just inter-mingle them.  

And this is where you show up.  And see a bunch of neutrals and grounds spammed onto the exact same bar, and go "WTH? Why 2 wires to the same thing?"  The whole logic of how we got here is not apparent to you, and it just looks redundant.  

Well, you can do your part to fix that. Go out and get accessory ground bars for your panel.  Move all your grounds to the ground bar.  (and while you're at it, fix any illegal double-taps, especially the deadly "put a circuit's N and G under the same screw" nightmare... because if that loses connection, you have the bootleg scenario below).  For more style points, remove the bonding screw/strap and replace it with a nice big fat copper wire between N bus and G bus.  Route it so you can put a clamp ammeter around it (why would you want to do that, haha! ;) 

Now we can return to our assumption that neutral is relatively safe.  

Fault current return

The other thing is, electricity travels in loops.  Power doesn't want to get back to ground, it wants to get back to source.  Source being e.g. the neutral at the transformer.  

So suppose our safety grounding actually worked, and it caught a bolted ground fault.  Do we want that fault to just be "caught" indefinitely?  No, we want a breaker trip to occur, and that means we need to flow enough current back to source (neutral) to trip the breaker for sure.  Again, the neutral-ground  equipotential bond gives us that path: Down the ground wire, over the N-G bond, and over the neutral bus back to the transformer.  A satisfying 500A of bolted fault flows, and the breaker goes SNAP! 

Or if we're in GFCI-land, a satisfying 0.007 amps of current flows, and the GFCI goes "Unequal current! Snap!" 

Let's bootleg ground everywhere! What could go wrong?

When you say "Hey, we don't need 2 wires", and just attach ground to neutral since they're the same thing in the panel, that is called bootlegging ground. 

Now, in ranges and dryers, appliance manufacturers got a special exception cut so they wouldn't lose appliance sales to people who had 3-prong dryer/range outlets.  They said "For grounding the dryer and range, let us bootleg ground legally. The socket is behind equipment and is rarely disturbed."  

Of course you can do the same at any receptacle you please, just by jumpering ground to neutral.  

So what happens if the neutral wire breaks?   Think about it. 

Normally, the appliance quits, but its conductance pulls the neutral wire up to 120V, since the power can't get back.  

Since we bootlegged ground, the ground is also pulled up to 120V.  We have electrified the chassis of the appliance.  And yeah, this has killed people.  It's always written up as "defective wiring", implying it was wired wrong... and that's true, except in the case of dryers and ranges which were legally bootlegged.  There, the wiring was "correct" by Code (certainly not correct in any real world evaluation), and a simple wire failure caused it. 

The electrical system ought to endure simple failures like a loose wire without it becoming hazardous.