Some experts say Mullah Haybattulah Akhundzada is likely to pursue aggressive attacks throughout the summer.
The Taliban leader was overseeing plans for new attacks on American targets, U.S. officials said on Monday.
Prime Minister Nawaz Sharif, who confirms Pakistan didn't have advance warning of strike, calls operation a 'violation of Pakistan's sovereignty.'
Mullah Akhtar Muhammad Mansour is hoping to end months of infighting.
Militants breached the heavily-fortified airport, which serves as a base for Afghan and Nato troops.
The clashes follow days of confusion over the fate of Mullah Mansour.
Reports said Mullah Akhtar Mansour was wounded or killed in a shootout in Pakistan this week.
An American-backed reconciliation process in Afghanistan has been cast into doubt.