ISLAMABAD—The rise of Mullah Fazlullah as the Pakistani Taliban's new leader marks not only a power shift within the militant network, but also threatens to ignite fresh conflict between Islamabad and Kabul.

Earlier this month, a U.S. drone strike killed the Pakistani Taliban's chief, Hakimullah Mehsud, prompting a power struggle within the movement's ranks. For the first time in its six-year history, the group tapped a commander from outside its cradle in the North Waziristan tribal area, selecting a native of Pakistan's...