Since Gen. Choi's Taekwondo was designed for the military, the round kick was designed to be used while wearing combat boots, so the toes did not have to be curled back. Instead, the point of the shoe was purposed used. With the full pivot of the body, swinging of the leg, and chambering of the lower leg backward, and then snapping it out, and back to the chamber position in a whip-cracking motion , much power is imparted to the small impact point at the point of the boot. When kicking barefoot, the toes must be curled back so the impact point is the ball of the foot. After rise of the Olympic sport style of Taekwondo, there were a lot of complaints about the "slow speed" of the round kick and toe injuries from kicking training bags with the ball of the foot, so a different type of round kick was developed. A knee whip motion with an impact that pushes beyond the target was developed. This was further aided by the in-stepping and kicking at 45 degree angle. This 45 degree round kick was used in the traditional Taekwondo, as in the Choong Moo pattern, but was not used much in sparring.
Therefore, there are two types of round kicks in Taekwondo. The old, traditional style and the new, modern style (bit chagi). Both of which are different from the karate "mawashi geri" roundhouse kick.