Comparison between the School Kyokushinkai Karate and Shotokan Karate Schools. For more realistic and objective we will take all aspects of both styles point by point.

In general comparison between Karate Shotokan against Kyo Kushin Kai Karate:
Both styles of Karate are styles of Japanese martial arts and mat be considered that each style is a different independent martial arts almost.
Shotokan is a straight style that tends to rely more on strength and speed than skillful deflection. Their training is intense but they tend to favor kata over sparring. Their gradings I know nothing about because I've never been to one. Their tournaments tend to offer three formats, Kata, semi contact point sparring and continuous gloved full contact sparring with minor points scored for throws sweeps, punches and kicks and major points scored for knockdowns and knockouts.
It has a heavy Japanese influence. 
Lyoto Machida is a Shotokan student.

Kyokushin is a mixture of the two and the newest. It is a point and circle style meaning that students are taught the direct Shotokan path first then start practicing the more skillful Goju-ryu style deflections, it also uses the Muay Thai low kicks. Training is a heavy push towards fighting and perseverance. Gradings are generally a test of stamina, ability and sparring. The shodan grading is at the very least warm-ups and basic technique with the lower grades, a five km run holding a punching bag over your head, kata with the lower grades, a breaking demonstration, holding the bo out for fifteen minutes, running end to end of the hall doing push-ups, sit-ups, burpies and squats, Kata with the higher grades, Sanchin kata while a higher grade hits them and forty rounds of Kumite. Kyokushin tournaments have four formats Kata, non-contact point sparring, K-1 rules matches and the full-contact bare-knuckle tournaments that they're famous for.
Bas Rutten and Georges St Pierre use Kyokushin as their stand-up style.
K-1 has always been chocked full of Kyokushin students.

If you still do not know what is best style for your child then read MA pros and cons.
In depth Comparison of Shotokan and Kyokushin Karate:
Main difference are in Speed ​​and dodgy of Shotokan and Kyokushinkai:
The speed, lightness and dodgy is the most important requirements for player of martial arts Kyokushinkai speed and dodgy is a few somewhat, but some of the players has super speed in Performing strikes Shotokan characterized by speed, evasion and the sudden onset, these properties make shotokan is more effective in real fighting.

Strength and rigidity in Karate Shotokan and Karate Kyokushinkai :
Weak player will face a lot of difficulties while fighting against the largest and most powerful of him, therefore power is also one of the strongest requirements of martial arts
Kyokushinkai depends mainly on the strength and hardness,it beats the Karate ten times in the power of endurance!!

Shotokan has one of the biggest disadvantages which is lack of sufficient rigidity during the fighting sessions and this is something that has helped in joining the International Olympics after being ruled out in after Olympic's new law which excluded shotokan.

Differences in Techniques and skills of Kyokushinkai and Shotokan Karate:
Martial arts Techniques are a set of movements that knows the fighter to reach full strong movement.
Kyokushinkai is often lack the ideal technique because most strikes is performed in serial striking technique by hands then followed with kicks on lower leg or thigh (or vice versa)
Karate Shotokan is characterized by abundance of techniques.

Default Style of the game or comprehensiveness:
Differences in Punches, Blocks, and kicks of Karate Shotokan against Kyokushinkai:
Any of the martial arts is not complete. Most martial arts lacks one aspect or another of fighting skills. for example Judo player misses a beating skills and player of taekwondo misses skills of throwing and so on.

Kyokushinkai aim on prevent the punches on the face, hands makes the player of Kyokushin (in the street) more vulnerable to injury,his hands that do not protect him enough and also his familiarity with the style of fighting Kyokushinkai also does not protect his face as required, however, this lack of face protection technique may be offset by the rigidity of his body and resilience.
Shotokan: the most comprehensive in the strikes, protection and style points it makes Shotokan Karate player counter intuitive faster and faster in strikes too.
Image show high kick of both styles of karate.
Differences in Strikes of karate shotokan and Kyokushin kai:
Kyokushinkai strikes are very strong especially strong kicks, but the style of Arbitration of game leagues judging to win the player most strikes and this makes some strikes especially targeting in the stomach are just to little increase in points, but at the same time be strengthening exercises and hardness of the other player.

Strikes in Shotokan are weak and very limited (in official matches) and this is the outlet of many on this game but the way it blows and the diversity of skills than game Kyokushinkai much skilled player of Shotokan which is rich of a lot of smack punches and kicks that are repayable on sandbags which are special exercises in addition to his normal training.

Differences in matches, leagues, Point system between Kyokushinkai and Shotokan.
Matches of Kyokushin kai and league is resolved rounds either by points or knockout and duration of 5 minutes per round, or four (I'm not sure) and after each round there is voting, if is not whipping the flags of the parties for more than one rule held a second round.

Matches of Karate Shotokan are less the trouble with that the judgment stop the players after every shot and see the rest of the referees If there is a calculation of the point is the calculation by the vote, for example, if the voice of judgment the arena point (Zhangjiakou Zucchi) for the red and was one of the referees had not seen the point, and the second was not charged by the third lineage of player the other rule, the vote is the most powerful scene is. And be the match of the round one period of 3 minutes and ended in a draw if there will be another round of one-minute (Anchosn) and if it did not win any player be any point the final thing is to vote.

Conclusion the main differences between Kyokushin and Shotokan are:
The fighting stance in Shotokan tends to be more traditional and low, while Kyokushin use the traditional stances to train in but fight in a Muay Thai stance that is higher than the Shotokan
The training in Kyokushin dojos tend to be more intense with lots of body conditioning and bag work, while the Shotokan places I trained in have more focus on karate history and meditation than Kyokushin. Shotokan tournaments are semi-contact or point or non-contact, Kyokushin has full contact tournaments. The Kyokushin kata is different as it takes influence from Goju karate as well as Shotokan. Karate students have held at least four UFC championship belts. most styles of karate use sparring gears in tournaments.