What's the difference between Haiashi-Ha Shito-Ryu and traditional Shito-Ryu? What does Hayashi-Ha have that Shito-Ryu don't have? And what does Shito-Ryu have that Hayashi-Ha does not?
Shito-ryu isn't exactly traditional. It was created by Mabuni during the 1920s in Japan. He based it on a formalisation of Okinawan kata. Mabuni was a kata collector, even more so than Funakoshi. If you do any shito-ryu, Haiashi-Ha or not, you will be learning a lot of kata.
Haiashi-Ha is a more recent organisation still and could be thought of as the more sporting version of Shito-ryu.
You could say Haiashi-Ha is to Shito-ryu what Shotokan is to Shotokai. It's more sporty, competitive, less application based and very much more popular because it was spread widely. If you do Haiashi-Ha, you will probably be doing more competition based practice. if you do the "traditional" shito-ryu it's likely you'll be taught more application based practice, but the more traditional schools may be harder to find.
What does each have that the other does not... Nothing in a practical sense for a student. It's all potentially there. The differences are a matter of emphasis, and like all styles it's going to depend very much more on an individual sensei than on anything else.
I follow Hayashi ha shito ryu and did classical shito ryu for a long time. Sensei Hayashi studied with Mabuni himself, but also went to Okinawa to study traditional styles. He studied matsubayashi shorin ryu under Shoshin Nagamine and also learned ryuei ryu directly from the Nakaima family. In Hayashi ha we have many katas from the shorin ryu school, which are different from what the Mabuni shito ryu has; ,examples would be Rohai and Chatan Yara Kushanku which are very different to the shitokai versions. Our Matsumura Bassai is shorin ryu based as well, with notable differences to the shito kai version. There's also specific information about hayashi ha shito ryu that you'll likely find if you attend one 8f their schools.