The opportunity to stay with a Japanese host family is the perfect way to immerse yourself in the culture, and practice at "home" what you learned at school that day. Host families are chosen to be as close to the school, and the center of Fukuoka City, as possible. You are guaranteed to have your own private room, which may be either Japanese or Western style.
A private room in a dormitory with other students and Japanese university students gives you the freedom to enjoy the city life without a curfew, but also the chance to make lots of friends.
Staying with other people in the city gives you the opportunity to make friends easily, and go out freely, but still have a good level of privacy and freedom. In a shared apartment you have your own private room, but share the central facilities, such as laundry, kitchen, living room, bathroom.