Main story will be better understood in that game itself. It's just a outcomes of a few questline that is teased in HTA and LSH.
Let me explain with a few examples.
- Toilet Ghost Hana- As you may know, you can get a secret animation with a ghost in girls toilet in college map. In Magic Ring, you will hear about this ghost after Mariam's(Momoe) death is murdered by her. But if you have played this game, you can try that thing before that event happens thus avoiding Mariam's death.
- Faith... You see her standing outside of the room where Mason and past Judy are having sex. When you play Magic Ring and leave the room to interact with her changing her decisions.
- You learn that Dahlia is killed by director of Hillford asylum. You can prevent her death in Magic Ring by not letting her investigate stuff on her own.
- Then there's this sales lady you find with Mason in apartment 202. From there conversation you will know you have to read the full terms and condition when a window appears in order to get a special option.
- Then there's one more information you get in this game that can be used in Magic Ring to change how one character interacts with you. This one will be kept a secret until the game is out.