I don't think we would have had a War on Terrorism, at least not right away We would have had some tighter security restrinctions implanted, but not nearly to the extent we had post-9/11. We probably would not have gone into Afghanistan. And, without a 9/11, it would be much harder to make the case for a War in Iraq. So we could see a Middle East where both the Taliban and Saddam are still in power.
However, this all assumes that will not be any future attempts after this, which I believe would not happen. This wasn't the first attempt of terrorism against the United States. After the Bojinka Plot failed, they did not quit. Rather, the lessons they learned from the failed plot helped them plan 9/11. So I think we could see a future attempt, possibly successful, with many of the masterminds behind 9/11.
But with no terrorist attack, or even just a later one, we could see Bush having a more developed domestic policy. In OTL, "No Child Left Behind" was one of his only major domestic policies successes. Without 9/11, we could see him attempt some more projects. Although, on the other hand, without 9/11, there's no guarentee that the Republicans will win the 2002 midterms (although I would not rule it out completely). Therefore, there's a good chance Bush will face a Democratic Congress through all of his first term. And the chances of him passes, say, Social Security privitazation are rather low. Plus, without 9/11, he possesses a greater chance of being defeated in 2004. But that's not a guarentee. If a later 9/11 happens, and he it happens during late 2003 or early 2004, that nearly guarentees him relection. Barring the public does not blame him for the attack, which doesn't seem that likely.
So, anyway, the path of this timeline mostly depends on whether or not there is a later terrorist attack.