Major differences between Hardtail and Full Suspension mtn bikes
It is as simple as full suspension feels like suspension on both wheels and hard tail feels like the front only.  

Use the full suspension for more and bigger bumps.
Also for landing jumps.  

The hard tail is lighter and the pedal action is more efficient.
The frame is stiffer laterally (more efficient).
The rear shock on a full suspension will absorb part of the pedal energy (very little as it is designed to minimize this).
For lighter bumps.
The tail will get bounced around (much) more than full suspension.
Bouncing a real wheel is typically not going to drop you but you are not getting traction.  

You will typically see downhill riders on full suspension.

On single track it goes both ways.
Your preference and the nature of the single track.

Trials riders will ride hardtail.

Some tricks like a bunny hop are easier on a hardtail (I think). 

If you are learning technique then I would go hard tail as it more forces you to pick a line.  

If you want to bang through some ruff stuff then full suspension.  

Tend to take harder falls on a full suspension as you can go faster on the bumps.
And when those bumps are rocks it hurts.
And when you ding a $1500 frame it stings.  

If you are going to get full suspension then get a good one.  

In looking at cost and weight of FRAME ONLY.
I am going to compare Niner as that is what I ride and what I know best.
Hard tail frame: Ross 9 large frame weighs 1,680g and retail list for $899
Soft tail frame: Jet 9 large frame weighs 2,980 and retail list is $1849
Now those are higher end frames and the difference is $1000 and 3 lbs
On the mid range the difference might be $400
Don't buy a low end full suspension!    

My two cents:
Don't buy full suspension for your first mountain bike
Learn technique on a hard tail
There are lots of times when you going to want to ride a hard tail
Make it your second bike and be ready to spend some coin
The guys and girls riding a full suspension for what it is are good and are on $3000+ bikes  

My three cents:
Upgrade your hard tail to tubeless and go 2.25 (or 2.4 in the rear if you can) in the rear
Not the same but at lower pressure you get less bounce (for NO weight)
And you get the grip of conforming to rocks and roots
And you get no pinch flats
For less weight than the full suspension