So I take it that the hijackers were caught? I'm guessing in this ATL the CIA and FBI and whatnot are more cooperative.
I don't think we'd be able to go into Iraq in this ATL, if Clinton couldn't do it with his "evidence", then the Bush in this ATL probably wouldn't be able to either. Bush changed after 9/11, so without it he'd probably just do the usual "try to get the UN to do something about the WMDs" routine/song and dance. So Saddam would remain the "Boogeyman of the Middle East" that he was during the 90's. Bin Laden would still be the "religious weirdo nut out in the deserts who wants to kill America", and still seen as harmless by everyone outside of the various intelligence agencies.
China would probably be the biggest concern, rivalled only by the problems in the Middle East. Just because there's no 9/11 doesn't mean those issues disappear.
Anyway, that aside, I really don't know who the Democrats would, in a world where 9/11 never happened, elect to run against Bush in 2004.
-Al Gore wouldn't be chosen, he came with too much baggage from the contested 2000 election. Plus he was pretty content OTL to follow his other interests.
-Don't think Hillary would run, I think she was still wanting to take a slight break from intense politics. Plus her running would polarize the Republicans against her due to her last name.
-Dean was running on an anti-war platform OTL, so without that, Dean really has nothing. So he'd probably not have a chance. And even with that platform in OTL, he botched the huge lead he had.
-They'd probably go with Kerry, he had the "War Hero" thing going, (though that didn't help McCain) so he's a possibility, but I doubt people would overlook his activities during Vietnam like in OTL. Would they make much of a difference in this ATL is the big question.
-Joe Lieberman running would split the entertainment industry, (some of the biggest fundraisers for the Democratic Party, look at how they were in the '04 and '08 elections) due to his activities in regards to video games, music, movies, etc. So he'd be too risky to be the main ticket headliner. Plus he was connected with the 2000 election, too much baggage.
So I'm not really seeing much hope for a one-term Bush presidency in this ATL. 9/11 contributed to alot of the economic problems of the day (and probably still some now), so I don't know what kind of economy there'd be in a world where the Trade Towers are still standing. There were problems being talked about before 9/11, but it didn't seem like it wasn't manageable. So.... besides those who were still bitter over the 2000 Election, I really don't see much in the way of strong Democratic Candidates.