Okay, sooooo...it's been a month, and nobody seems to have any clarity on this, or interest in answering. As such, I've done some research in the off time, so will take a swing at it. I've actually tried to find logical ways to biologically verify Super Saiyan transformation and Kaio-ken as two distinct states, while also showing how they are different.
My theory: While both Super Saiyan (SS) transformations and the Kiao-Ken technique (KK) basically serve to increase Ki within the body of the user, and both require intense preparatory physical training to enable the user's system to enter and sustain the states, I think the two boosted states vary in specific quantitative effects on the body. The KK increase acts as a singular, massive dump of energy into the user's existing physical structure, as opposed to the SS state using the energy to enhance that structure and push the body to a new form. By this, I mean while the overall effects seem similar, the "mechanics" of each practice differs.
If we were to compare it to car mechanics, the KK is essentially like putting a Nitrus injector into an existing engine; random blasts would allow for incredible bursts of speed and power. Concurrently, the SS transformation is more like upgrading that engine (Heart, body, nerves & muscles) to turbo, meaning overall horsepower would increase. Changing the quality of the Ki (fuel), which may occur during SS, may possibly be like adding super fuel as a tank standard. So it's likely, even having a basic premise, the effects overall on the body are quite different.
Arguably, from what we have seen, Kaio Ken (KK) amplifies the latent Ki in the body in amount, by basically doing something similar to unlocking the "inner gates" and letting it flow at 2, 3, 4, 5,10 or whatever times the normal amount of Ki one normally uses. This extra Ki could be pulled into the body... or (more likely) perhaps greater reserves of unrecognized Chi/strength were built up unconsciously by the user via the intense, limit-expanding training they endured to learn the KK. It's releasing that Ki flood or "spiking the adrenaline" which pushes the user's physical limits. This is backed up (somewhat) in the manga translation below.
In this regard, we can think of the KK as an "ki-energy adrenaline shot"; it's an optimizer which unlocks otherwise stored potential, and unleashes it in the body via a massive wave.
Real Life Phenomenon-- Adrenal adaptation
The analogy isn't so far off; in real life, trained athletes--- and more on the nose, skilled fighters--- tend to tap into their adrenaline naturally during an event. This can have several notable effects, including temporarily boosted strength, heightened sensory awareness and reaction time, decreased sensitivity to pain, and even increased aggression... all things we see Goku use in KK mode. However, too much adrenaline in the body can actually have negative effects, which is why your body doesn't release it in heavy doses unless its a Fight or Flight scenario. People high on adrenaline tend to show visible signs (shaking, flushing, pupil dilatation, etc), and although their focus is heightened, they don't "control" the state so much as are subject to it...and that's not even accounting for long term effects (ulcers, hypertension, DNA damage, et al).
The short-term consequences of soreness, fatigue, inflammation and crashing are very much like how Goku's body reacted to each of his Kaio-ken stages, due to the stress put on it. It could also speaks to King Kai's "training for various KK levels" idea, in that the training not only refined his Ki control, but developed Goku's body enough so that he could use these "bursts" without damage. Humans only actively use about 20-30% of our muscles' full capacity on average; if we used any more sheer force at any one time----i.e., the amount of strength used during an adrenaline rush--- the muscles could tear right off the bone. The obvious corollary would probably be a KK-user needing to "control" their Ki the way us normal humans must consciously control our bodies during exercise to ensure we don't over push, lock joints, swing the weight, overextend a punch, tear a muscle or otherwise hurt ourselves.
It's only through repeated conditioning and focusing/ visualization training that the power of adrenaline can be harnessed in the body of a normal fighter. Too much adrenaline in the body, however, can be dangerous in real life; it can over stress the tissues, cause the body to go into cardiac arrest, or even rapture blood vessels. I'm guessing this is very much like the KK, just with more obvious tissue damage and heat release [Ki is a radiant energy, after all]. Hence the need to have decent Ki control.
Human bodies which have developed this control----- or have been put through constant conditioning via exercise and thus adapted to periodic exposures of higher levels of stress------ tend to be acclimated to such high adrenal release. Naturally, they also tend to consequently be stronger, fitter, more hormonally balanced, and thus better able to cope with the stress of a sudden adrenaline surge. This is due --- and this is where the real world application of cellular alteration and epigenetics come along---- to the fact that the tissues in the body change as a natural consequence of exercise or repeated stress; everything becomes more efficient. The nervous system primes easier, muscles react more quickly, and negative adrenal effects on the body aren't so pronounced. Athletes with these traits---those who train regularly, and thus whose bodies are used to big efforts--- can then use this adrenaline spike to boost their in-the moment performance. And this is where the "upgrade" difference lays in SS compare to KK.
Super Sayain Transformations
SS transformations, however, while also utilizing Ki as the basis, are fundamentally different by means of being a "Full systems upgrade." For one, the effect is not "Short term"; all SS variants are essentially Ki boosts which cause epigentic changes to the user, permanently altering the tissues of the body on a cellular, or perhaps even molecular level. These tissues themselves are thus amplified by increasing both the amount of ki that flows through them, as well as their capacity to withstand strain and produce (or at least, to better utilize) the raw power. Ki may be the main power source, but it's the physical aspects of the user which are all upgraded; disregarding the change in art style over the years, a SS user's body gets noticeably more toned or different in size and definition. In real life, greater muscle size and striation and loss of fat are visual indicators of improved strength and cardiovascular conditioning.
As much of the training takes place under heavier gravity, we can presume all tissues (muscle, bone, organ) become denser, thus stronger and sturdier. Even without the gravity component, SS transformations seem to change the user on a genetic level. This is supported by the fact that hair color changes (a physical, cellular alteration) when SS states are induced. This would follow the pattern of all major Saiyan transformations, from SS to Oozaru, which cause various levels of physical mutations (or rather, the activation and expression of key genetic variables) which change not only how a Saiyan's body may function, but how the individual even looks outwardly. A clear example of genetic alteration would be going "great ape"; even with the genes there, when its unexpressed, a Saiyan looks basically human. The genetic activation is further evidenced by the fact that SS parents seem to pass these active genes on to their children (Goku to Goten.)
The above mentioned changes could be seen as epigentic expression, but also when Saiyains hit a certain SS level, the bodies' very muscles themselves are holding so much power that they expand in size. I'm guessing what happens is that the mitochondria (the powerhouses of a cell) are packed more densely with bio-energy, so the cells themselves bulk up to hold the energy much like a primed muscle holds a "pump" after exercise. From here, muscles do what they normally do; adapt to the new strain placed on them by becoming more efficient, and using the greater amount /purer type of bio-energy or Ki to do the same amount of work with less energy output or effort. Simplified: even if not "looking" like it, the muscles are stronger and more efficient, doing more work with less recruitment. Hence why upon reaching higher SS forms, the Saiyan's base form becomes stronger than it was previously.
So the main difference then, as I see it is: going Super Saiyan effects a more permanent change to a Saiyan's body, possibly on a genetic level, altering their nerves, bone, organ systems and of course muscles so that they can optimize the energy, thus setting new upper limits to the body. Going Kaio-Ken, in contrast, simply spills more available Ki into their existing form and allows it to surpass its established limits, ....but only momentarily. This would explain why a Saiyan must hit a certain physical power level before Ki can multiply itself.
However, due to the nature of how DBZ portrays the relation of Ki to strength and Ki being dependent on the physical state of the users, I'm still unsure if it's the ki that allows them to affect the change.... or if the change allows a SS to generate more Ki. I'm guessing it's the former, because we've steadily seen Goku's energy reserves grow and him use ever more-absurd amounts of the stuff with each new SS level he ascends to.