I was wondering if anyone would be interested in giving their thoughts/opinion on this topic. I am curious to see if anyone thinks about this in their training, if anyone knows why the differences exist, and maybe any personal opinions on the pros/cons of each or how your own mechanics look when striking. I don't think this is a topic that is very commonly spoken about, but I have been looking into this recently and thought I might make a post about it on here.
During my time spent in the martial arts so far I have been exposed to throwing punches the Karate way as well as the combat sports (Muay Thai/Boxing) way. One of the key differences between these methods of punching is the shoulder mechanics. In Karate punches, the shoulder is often kept "down," like it is for all other basic techniques in kihon and kata. On the other hand, combat sports style punches involve a certain amount of movement of the shoulder. Not only does more rotation and free forward movement occur, but the shoulder can also be raised thereby covering the chin during the punch. This is a very simplified explanation, but hopefully you get the idea.
In terms of how this relates to generating power, I think that the Karate punch is more suited to "driving" the shot into your target, and is a bit more rigid structurally. On the other hand, the combat sports punch is less about "driving", and the shoulder mechanics allow power to be transferred from the ground up, almost as if you are "throwing" your fist at the target. This explanation makes more sense when you look at the follow-through of the different types of punches. Combat sports punches tend to be "thrown" and follow through the target much more freely, whereas Karate punches go through the target, stop at an abrupt point, and then snap back. Difficult to put into words, but hopefully you get the idea.
With all of this in mind, I also believe that one can throw punches with mechanics that are somewhere in-between those two methods. For example, I remember being told by a Karate instructor (somewhat early on in my training) that I could "let the shoulder go a bit" while throwing punches from a fight stance. The end result was somewhere inbetween the two, and might work as an effective compromise in my opinion.