On the day, the long war between the Leaf and the Cloud Village ended, and a peace treaty was made. Someone tried to kidnap Hinata and steal the Byakugan's secrets. The kidnapper was spotted, and easily taken care of by Hiashi. The kidnapper was the Cloud Village's head ninja. It's then when The Cloud accused the Leaf from breaking the peace treaty, and demanded Hiashi's body. It's obvious that the Cloud went to the Leaf Village with the intent to steal and reveal the secrets of the Byakugan all along.
But the Leaf, not wanting to risk another war, didn't decline their demand right away, and hosted a meeting with Hyuuga. Hiashi agreed to the matter, and was willing to die for the sake of the Leaf Village. But just when he was about to, Hizashi interfered.
Hizashi wanted to go instead, so that the Byakugan's secrets wouldn't be revealed. (The curse seal casted upon all Branch House members seals the Byakugan ability upon death.) But the real reason why he wanted to give his life, wasn't because he was a branch family member (protectors of the main house), and Hiashi was a member of the main house. No, he hated and detested the main house of Hyuuga. He did it to protect Neji, his family, and his village, ...to save his brother. Just once, he wanted to go against the destiny of Hyuuga. He said he just got urge to choose his own destiny, make his own decision and choose his own death. To choose freedom, that's all.
Ever since that event Hiashi has felt guilty for the death of his younger brother. He only tells Neji what really happened to his father when he was old enough to understand. This was right after Neji lost the fight against Naruto in the chuunin exam. Hiashi gave Neji a letter written by Hizashi, addressed to his son. He then kneels down before Neji and begged for forgiveness.