So first off, this is a long shot:
- US intelligence at the time was much better about understanding and targeting neo-Nazi/white-supermacist plots, albeit in part because there was a critical lack of understanding of how bin Laden operated and how his PR worked.
- Most of the US white-supremacist organizations were already known and monitored; al Qaeda wasn't.
- Suicide hijackings aren't, as has been mentioned, the neo-Nazi style. More likely they pull a mass shooting at a synagogue, blow up a black preschool, or firebomb a government building.
Let's handwave the first two; a white supremacist prison gang, further radicalized by the McVeigh bombing and the Waco and Ruby Ridge incidents, manages to concoct a plan with some KKK goons who want to do more than bitch about the "uppity n***ers" and wear stupid sheets. They smuggle a bomb into DC, but fail to hit the White House; instead it blows up in the street or something. Simultaneously, they drive a truck bomb into Howard University, killing dozens to hundreds. An attempt to shoot up a synagogue is mostly foiled by fast-reacting worshipers and a conveniently close police car. Total casualties, a few hundred people.
There is going to be a PATRIOT act with more bipartisan support and even more domestic surveillance. Iraq is not likely to happen, at least not as soon as it did OTL; it's going to be a lot harder to justify going in to the public when the zeitgeist is more concerned with domestic extremists than with foreign insurgents. The neo-Nazis basically cease to be a threat by the mid-2000s as public opinion turns harshly against them and they can't sustain recruitment.
Jack Bauer fights insidious neo-Nazis who hate America because we love our freedom. He still becomes controversial because of torture scenes and the like.
Both parties likely drift left on social issues like affirmative action, LGBT rights, et cetera, but also become more authoritarian. This lasts half a decade or so, then around '06 or so criticism from antiestablishment types in both parties begins to challenge the prevailing security-state center. Ron Paul and Bernie Sanders or Paul Wellstone (if he lives) will likely run significant '08 antiestablishment campaigns as the immediate terrorist threat wears off and the economy tanks. However, without Iraq, it's likely that the economy will be a bit more stable going in to '08, and the general sense of malaise may be less because of fewer unpopular foreign entanglements. I'd expect an antiestablishment coalition of the progressive/proto-socialist left and the libertarian right, united on opposition to security measures. It probably falls apart by 2012-2015.
Any further predictions run the risk of falling into current politics IMO.