Kyokushin has a lot more focus on hard sparring. As Dave writes in his answer, for tournaments, Shotokan fights are usually stopped after a successful technique lands (much like fencing), while Kyokushin fights are only stopped when the technique is effective (i.e. the opponent is either knocked out or knocked down).
Knockouts are common in Kyokushin, while excessive contact is penalized in point karate.
As for the kata, I believe Kyokushin and Shotokan share many kata, although their interpretation may be slightly different. However, depending on the school, it might be possible to focus almost entirely on fighting and less on kata (or vice versa) once you reach a certain level.
If you enjoy hard sparring but not the kata, you might consider one of the Kyokushin offshoots.
Ashihara Karate is a direct offshoot of Kyokushin, and shares the same view on hard sparring (knockdown karate). However, their view on kata is different. While traditional kata are very stylistic, the Ashihara kata are more practical. The intention is, that the individual combinations can be transferred directly into fighting.
Example of Ashihara kata
Enshin Karate is in turn an offshoot of Ashihara, and is very, very similar. The only difference is that they have fewer kata.
There are other styles as well: Daido Juku/Kudo, Seidokaikan (parent organization of K1) and several more. Most of them are descendants of Kyokushin or Ashihara, and are grouped together as "Knockdown Karate".