Hi guys, I'm the offending commenter that Tennesseejoe was nice enough not to mention by name.

No worries, Joe, it's definitely an important idea worth discussing separately from the pushout thread.
I understand why pocket game players have an irrational love of stun caroms since I grew up playing it before discovering billiards. We're constantly taught tangent lines, kiss shot caroms, etc, and straight lines are so so simple to think about...how could anything be easier, especially where there's mysterious curving involved? Well, as they say, the devil is in the details...
Each of these two "types" of carom have advantages over the other.
I think the only advantage stun caroms have over natural roll (NR) caroms is you can reach angles greater than the NR maximum. And I say this as someone who plays stun caroms very well, so don't think I'm just ranting on about them because I suck at them. They're really fundamentally less accurate than NR caroms, and that's why billiard players only resort to stun when NR isn't an option. Of course, some positions you need to do something different with the object ball that disallows NR, but we're talking pure carom accuracy here. I'll address why below.
Stun
Pro: The CB moves in a straight line at exactly 90 degrees (easy to visualize) to the objects ball. The direction of the CB does not depend at all on the speed of the shot, as long as the CB is sliding at contact.
These are commonly taught principles, but neither is really true. To know why, see Dr. Dave's paper
TP A.5 - The effects of ball inelasticity and friction on the 90° rule. Ball inelasticity and friction ensure that your carom line will always be less than 90 degrees, and both coefficients depend on shot speed and cut angle.
Con: Getting true stun is often very touchy, especially if you have to hit the ball very hard which requires an extreme level of accuracy in your tip placement.
This is absolutely true, but most pool players underestimate the importance of this (apparently, even very knowledgeable ones like Andrew and Patrick - sorry guys, don't hate me

)
Everyone thinks they can stun accurately because the cue ball stops dead when they shoot. Actually, it's only stops "mostly dead" (and is therefore "slightly alive" ala
Princess Bride 
) with slight movement one way or another. This is especially true as shot distance increases (although being close has other issues, like jumping due to natural cue elevation.) Since the slight movement is usually irrelevant when playing position, it's promptly ignored by most players and accepted as perfect. It's
very difficult to get the instantaneous state of a purely sliding stun (where the carom is already off of 90 degrees by 3+ degrees as mentioned), and that "mostly dead" stun is worth at least another few degrees of error as well.
Some players choose (or intuit through trial and error) to err on the side of a little draw, which helps compensate for the inelasticity and frictional effects and puts the ball back closer to the 90deg line, but most players are just not getting the complete stop they think they are. It may seem insignificant, but that slight error gets magnified as your target distance increases. (For reference, 1/2-diamond across the table width is only about 7 degrees, and that's bigger than a pocket or 2 balls.)
Andrew made some good points about the pros of NR caroms, but I'd like to fill in some specifics:
For reference, a tip height of 2/5 radius above center gives the CB natural roll immediately. Anywhere around that height, and it will equalize to NR in a short distance, especially on anything but brand-new cloth. You can even play predictable NR caroms right next to the OB if you're careful about tip height. (I've found most pool players tend to not hit high enough when they first try this, though, and they don't get a full NR.)
For a NR hit between around 3/8 and 5/8 ball, the cut/stun angle varies about 20 degrees while the carom angle varies slightly less than 2 degrees. That means the 1/2-ball NR carom is 10x more forgiving of aiming errors! The additional stretch out to 1/4 and 3/4 or so adds roughly 10 degrees of cut/stun variation for only 5 degrees of NR variation. That's still 2x more forgiving than stun. In all cases, NR is less sensitive. (Draw is considerably *more* sensitive, but let's not go there just yet

)
As for the cons: Yes there is parabolic curving before the straight line happens, and yes it takes longer to finish the harder you hit it. But that's pretty irrelevant from a practical perspective unless you're curving around an obstacle, especially if it leads you to only playing NR caroms softly out of fear of losing control of the curving.
Your speed control on NR caroms will improve greatly if you ignore the curving bit and just think about speed shifting the straight part over from the known slow reference line. You'll learn your limits quickly like this, and calibrating speed this way is
much easier and more forgiving than mastering all the speed/draw combinations players use to achieve stun, especially when changing tables frequently like in tournaments. For one, you're only manipulating speed instead of both speed and spin. Also, slight speed errors don't shift a given NR track much at all, but they really exaggerate stun errors by causing slight follow or draw at OB contact.
Final thought: rolling is what the ball naturally wants to do, and it's always working towards that state. Learn to appreciate it and use it to your advantage. It's the only time you know *exactly* how much spin is on the ball. Any other state and the spin/speed ratio is constantly changing, and it changes by different rates on different equipment.
Robert