A lot could change, I think.
Al-Qaeda was founded as a result of a meeting between Bin Laden, a man named Abdullah Azzam and the leaders of the Egyptian Islamic Jihad in 1988. Azzam was Bin Laden's mentor and was one of the high profile people responsible for funding, recruiting and organizing the efforts of Arabs, coming to partake in the war in Afghanistan during the 70s'. Azzam did this by running a organization called Maktab al-Kadimat (or MAK) which worked with the Pakistani ISI to funnel money to the Afghan Mujahideen. One of MAK's largest fundraisers was Bin Laden.
Azzam was rivals with the leader of Egyptian Islamic Jihad and future Al-Qaeda second in command, Ayman al-Zwahiri. Azzam wanted to use MAK's funds to install a Islamic government in Afghanistan and viewed attacks against Islamic governments as 'fitna' (Translated from Arabic, fitna means 'sedition' or 'civil strife').
Zwahiri wanted to use MAK's funds to support a global jihad. In 1988, when AL-Qaeda was founded, an agreement was made to link Bin Laden's vast financial resources with the Islamic Jihad and take the jihadist cause elsewhere after the Soviets left Afghanistan, which happened the next year in 1989.
Without Bin Laden to act as a third party, it's possible that the rivalry continues and Al-Qaeda is never founded. Egyptian Islamic Jihad's targets included the Egyptian government and American and Israeli interests in Egypt, so they would never attack the US. It's likely that without Al-Qaeda, Ayman al-Zwahiri remains in Egypt to lead his terror group, or flees to Sudan. Counter-terrorism operations would focus on multiple jihadist groups, rather than one overwhelming group like Al-Qaeda. Maybe al-Zwahiri plots this timeline's version of 9/11 from Sudan, rather than Afghanistan, which is probably still caught in a civil war since the Taliban only succeeded in Afghanistan because of Bin Laden's money.
As for Azzam, I doubt he would become a terrorist himself, but he would definitely become someone sympathetic to Islamic terrorism. After Afghanistan, he preached about jihad, saying in one speech.
"Blood must flow. There must be widows; there must be orphans."
His trademark catchphrase was...
"Jihad and the rifle alone: no negotiations, no conferences and no dialogues."
In our timeline, he was assassinated in 1989 by a car bomb in Peshawar, Pakistan. No one knows who did the deed. Suspects included Bin Laden, Ayman al-Zwahiri, the CIA or Mossad. But, assuming he was assassinated by rivals in the jihadist community, I don't think this attempt would take place. Azzam would probably continue preaching jihad before meeting his end by either a Mossad bullet or an American drone strike.
So, with no Al-Qaeda, the War on Terror, as we see it, would either look different or would not happen at all. The American homeland might be safe from organized acts of terror, but it would still have to fear lone wolves, like Ramzi Yousef in OTL.