Gore proved pretty adept after 9/11 in arranging the alliance with Ahmad Shah Massoud to go after the Taliban and AQ (Thus keeping much of the effort locally based, reducing possible American losses), and basically developed the "Lighter is Better Doctrine" through focus on using Spec Ops along with airpower to assist Massoud in taking down OBL at Tora Bora. Would Bush be as skilled as doing that? OP seems to suggest he would've went full bore with an outright invasion, by I think Perhapsburg is more on the spot in his estimation of Bush. He (Bush) would probably follow a roughly similar course that Gore took, so as to keep US presence minimal.
So while the Anti-AQ campaign would probably remain basically the same as OTL, domestically a lot could change. While the military successes of late 2001/early 2002 definitely propelled the Democrats to the major gains they got in the midterms in '02, said successes ultimately doomed them as most of you probably know. By 2004, the conflict was starting to fade in public focus with vengeance having been dealt and as the full impact of the Dot Com boom continued to be felt (And in public perception, was amplified without foreign adventures to act as distractions). So as the poor economy became the main issue, you then had the added historic nature of J.C. Watts clinching the GOP nomination as the first Black candidate of a major political party added on top of what Gore had to face going into 2004. Given both of those facts, Gore's defeat was to be expected. However, with a Republican in office, Watts would be unlikely to get the nomination due to the incumbency advantage and it's likely Bush would've got the shaft for the poor economy. So who were the Democrats likely to run? Maybe Bill Richardson could run four years early? However, even with the poor economy, given the scandal plagued background of Richardson, Bush could still probably beat him easily (Although without the 350+ margin Watts achieved in '08).