The Hyūga Affair (日向の件, Hyūga no Ken) — as the incident is known nowadays in Kumogakure — happened nine years prior to the start of the series, while Kumogakure and Konohagakure were at war. In an attempt to put an end to the fighting, they agreed to a peace treaty to be signed on the date which coincided with Hinata Hyūga's third birthday. The treaty was soon revealed to be nothing more than a cover when the person who was the Head Ninja of Kumogakure at the time — who was in Konoha to sign the treaty — attempted to kidnap Hinata in order to obtain the Byakugan and its secrets. The shinobi was killed in his attempt by Hinata's father, Hiashi Hyūga.
Kumogakure denied all accusations of the kidnapping, and demanded the body of Hiashi as compensation for the death of their Head Ninja as per the stipulations of the treaty. Hizashi Hyūga decided to go in his brother's place as they were twins and Kumogakure would not be able to tell the difference. This had another advantage, besides saving the life of the head of the clan; Hizashi's cursed seal would seal his Byakugan upon his death, safeguarding its secrets. Despite Hiashi's protest and their father's behest, Hizashi chose on his own to do this, not only for the clan and village, but for his brother.
The details surrounding Hizashi's death are quite well-known among the high-ranking ninja of Konohagakure, most of them taking it as a serious infraction on Kumogakure's part. This was shown when Yamato called out the Fourth Raikage for his relentless pursuit of Sasuke Uchiha, comparing the latter's supposed abduction of Killer B to their attempted theft of the Byakugan and noting that Konohagakure remained magnanimous while Kumogakure were vindictive.
While they are not allies, Konohagakure and Kumogakure seem to have come to cordial terms in recent years.
Despite Neji never showing much love he rebuilt the clan with Rock Lee where they lived happily from then on