As someone who uses both punches interchangably I feel it would be beneficial to weigh in on the benefits of each.
- Corkscrewing it in allows for the muscles of the shoulder to add power to the punch which makes it very powerful at full extension.
- Because the elbow rolls up slightly it also allows you to roll your shoulder towards the neck and chin to protect the side you have left exposed while extending the hand (this is the main reason boxers do this punch as they are not as concerned about a punch to the ribs with 14 oz gloves as they are concerned about a knockout bunch to the button on the chin)
- Finally the slight raising of the elbow makes it easier to punch over top of the opponents gaurd as your arms are less likely to clash.
- The vertical punch generates full power at less than full extension because it is not relying on any corkscrewing to generate power. This makes it very effective for close range striking (That's why Bruce Lee's one inch punch was vertical)
- As mentioned on horizontal punch a vertical punch will clash with the opponents arms more often which in some arts (such as Kung Fu) is considered a benefit as ever block is treated as a punch and vice versa, hence if you use sticky hands vertical is the way to go.
- If you punch with the top two knuckles you can add a snap of the wrist into your vertical fist to add force over a very short distance.
I have also noticed that most people on this board are implying that a horizontal punch is always done with the first two knuckles and a vertical punch is always the last three. This isn't true. It is a matter of bone alignment, those arts which are punching with the first two knuckles are alligning the wrist to the radius and those that are punching with the bottom three are aligning with the ulna. I personally prefer the top two as I would not trust my little finger to survive any amount of force. In fact when I teach a hook punch I teach it with a vertical fist to avoid leading with the little finger. I heard a story once about Tyson breaking his little finger in a brawl because he threw a hook and without the protection of the wraps and the gloves it broke.
So the punch you choose will depend on your style and situation. I tend to use my horizontal punch at range because it better defends my chin and then the vertical punch when I zone in to short range. I punch with my top two knuckles but if you are doing sticky hands where you are rolling your punches in this is likely going to cause you to punch with the lower three.