The Rare and Excellent History of Saladin discusses this some with an obvious bias towards Saladin, and Saladin: And the Fall of the Kingdom of Jerusalem discusses it as well. The reasoning is Nur ad-Din is supposed to have begun to distrust Saladin after he so quickly secured his position in Egypt, and subsequently refused to offer military aid to Nur ad-Din on two occasions - once because he had reports that the Crusaders were working with Fatimid holdouts in Egypt to instigate a revolt, and another because Saladin's father was injured and subsequently died. The second incident certainly happened, but the question of whether Saladin used it as an excuse to let Nur ad-Din fail and actively sought to set himself up as an independent leader can't be definitively answered.

I disagree somewhat with how the relevant portion of the article is worded as well:

After Nur ad-Din's death, however, Saladin took the initiative; immediately striking into Syria, disinheriting Nur ad-Din's heirs and subjugating the entire region.

It's a bit disingenuous and oversimplified. Without getting too complicated as to the politics, large parts of Syria declared independence after Nur ad-Din's death, considering his heir As-Salih Ismail al-Malik was 11 at the time and under a series of regents. While Saladin did declare himself king of Syria outright eventually, it's not unreasonable given his actions before then that he was honestly attempting to recapture the independent Syrian cities in As-Salih's name.