Depite their similar names, slings and slingshots are entirely different devices. Slingshots are much more like bows in that they are energy storage devices. You draw back the pouch, aim and release. The slingshot does the work.  
 
Slinging is really improved throwing; the sling makes the action a lot more efficient than throwing by hand alone. But it's still very physical which I enjoy. Hitting the target is more difficult and challenging.
 
From a historical perspective, the slingshot is absolutely brand new. Prior to the availability of rubber, less than 200 years ago, there were no slingshots. Compare that with the thousands of years that slings have been around.
 
I guess the sealer is how much time I spend doing both activities. I sling almost every day. I've tried all sorts of sling lengths and types, and ammo combinations. Yet I don't get tired of it. On the other hand I haven't used my slingshot in several months.

As young boys we were never too far from a slingshot. The slingshot used to be an ionic part of boyhood in America. Dennis the Menace, the Little Rascals, Wile E. Coyote and countless others all employed the use of the forked stick and red rubber tire inner-tube. I was young boy scouting around in the barn for rubber when my dad whipped me up a sling and I've been slinging ever since. I was in my teens when my older brother came home wiht a Wrist Rocket. I still have my original slingshot. I've taken more small game with the slingshot and when chasing away stray dogs for example it's quicker and more accurate for short distances but like Aussie stated, I never get tired of it and it's my sling that goes with me everywhere and not the slingshot.

Given the choice I would rather be better with a sling than a slingshot.  
 
They both have their uses. As a long term survival weapon the sling wins hands down. Since that is what interests me that is what I would rather use. I feel like the sling is more fun to use than the slingshot as well.  
 

I voted for slings because I can make them myself out of many different materials.  I don't need a piece of some synthetic elastic material.  Also the sling is more challenging to use.  That said, if I was in a survival situation where I had to kill small game (birds, rodents, etc..) to survive I'd pick the sling shot because they're much more accurate.  Both are good weapons and have their place.

What about for driving away humans or large animals threatening you or your livestock (assuming the choice is limited to slingshot or sling)?

as aussie said. Totally different weapons.  
 
Slingshots are short range and can be aimed. They use light ammunition, and for the slingshot to be effective the ammunition needs to be quite dense. Uses stored energy
 
Slings are unaimable, multi-range, multi ammunition weapons and convert the body's movement into energy for the missile.
 
It really is just the name that joins the two.
 
It's exactly like comparing bows and atlatls.
Both use long pointed amunition, but that's about it.  
 
now comparing atlatl and slings - far more interesting.  
 

Quote from curious_aardvark on Apr 23^rd, 2012, 6:40am:

 
Sling vs atlatl is interesting.  If we're just talking a day out in the woods trying to secure game, I'd probably take an atlatl and about six darts over a sling.  My username sort of says it all.  It's a very familiar weapon for me.  The sling is getting there, but I'd still place a bet on my atlatl accuracy any day of the week.  If the game is a survival situation, I'd rather have the sling, because the ammo is freely available off the ground.  One of the reasons I train so hard with the sling is that it's like a pistol if bullets could be found paving roadways.