Here's how I reckon things would go under Gore and how that would influence pop culture:
9/11 still happens and Al Gore receives a huge boost in popularity and in political influence. As such, he would then target Al Queda in Afghanistan.
Would we have the Iraq War? Maybe, though if we did, the details would be much different.
Let's start with Afghanistan first. Al Gore would've retained the focus on Al-Queda and Bin Laden; let's say he manages to actually get Bin Laden in the first few years. This would boost his popularity even more. He would be out some moderate to liberal Supreme Court judges which would affect things in the long run. However, he would probably use his initial war boost to get some strong green energy policies in.
All of this would make him come off as this stoic but professional badass that was intelligent yet tough. South Park might still try and make fun of him for his green policies, but perhaps not as much given his success (and maybe it'd just get them more backlash.) He'd be viewed as efficient and thorough. It would also probably put the idea of "going green" more acceptable in the zeitgeist though I do not know the details of what Al Gore would do. Carbon credits could be something though if he had a lot more clout, we could see a carbon tax or at least green energy subsidies. Conservatives would try and make claims, but his success over Afghanistan would likely keep the noise down.
This leaves Iraq.
Al Gore has always had plans for Iraq and his criticisms seemed to be more aimed over at how Bush did it (being belligerent, not good with European allies, etc), but there is one potential thing that would toss Gore's target for Iraq for a loop.
Back in 2003, Iran actually went to Bush and proposed a deal that included things like ran offered a proposal for the US. From what I gathered, it'd be for the US to lift sanctions in return for nuclear cooperation, recognition of Israel, and an end to state-sanctioned support of groups like Hezbollah and Hamas. Bush initially said no and it'd bite him in the ass down the line and make things worse. Hard to say with Gore and it would all depend on if Bin Laden was caught and when.
There was apparently a chance to do so back in 2001 though he escaped. If he was captured then and there, it would give Gore a fair bit of clout and put through the procedures. Granted, the timing is fluid.
Back to the deal. I am thinking that ultimately Gore would take the deal. If he captured Bin Laden beforehand, he could weather the criticisms. If he did not, it could cause trouble unless he would also try and get Iran's help in taking out Bin Laden.
Assuming the former and if Gore has not invaded Iraq yet, the discussion of Iraq would definitely be on the table with Iran and beyond securing recognition of Israel and the end of backing with Hezbollah and Hamas, Iran would be interested in a deal involving taking out Saddam. They might try and claim the Shia-claimed areas at most, but I doubt they'd be that bold unless Bin Laden hasn't been caught, and even then, they'd likely have to give more tit for tat here.
Iraq would be a bit of a shocker there, but Gore could point to the various atrocities, such as the attacks on Kurdish Iraqis years prior and maybe he would try and let Iran claim some Shia Iraqi land in exchange for Kurdish territory, though no idea how Turkey would react. I also do not know if Europe would go along with Al Gore here. Assuming Europe does not go with, the Iraqi War may still be done though with Iran as potential assistance. This would give a bit of a colder ruthlessness to Gore that would appeal to the more conservative elements, but his more leftist supporters would find harder it a lot to swallow.
Like everything, it all depends on execution. Though the Iraq War may end sooner and rebuilding easier if the Shia-dominated parts of Iraq are granted to Iran and the UN persuaded to come in and assist with Kurdistan, which leaves a smaller Sunni Iraq. Iraq would very likely keep Baghdad as capital.
Assuming all of this happens before 2004, Al Gore would likely be reelected. Having captured Bin Laden early on and later helped bring better relations with Iran along with dealing with Iraq would make him a good bit more popular than Bush would.
Pop culture would have clean energy and other Gorean green policies be taken more seriously and probably more accepted into the general public. His works would definitely be more popular, to say the least. This drift of popularity would wane a bit until greater UN climate policies would be noted and appear to vindicate Al Gore.
Hurricane Katrina would be something of an odd bag. Al Gore could probably handle the whole thing and rebuilding better than Bush. He could get some flack for mentioning climate change as in relation though he would be vindicated of this down the line.
A fair bit of pop culture things would remain the same. FPS games remain murky. Most likely we would have them, but if Gore finishes the Iraqi War relatively quickly with Iran's help, we would get some games representing that. And for some reason, I could see a few FPS games made that have environmentalist elements to them (maybe someone makes an FPS game about potential Scarcity Wars?). So FPS games would likely not remain in the public conscious as long and would more quickly turn to more fantastic or differing elements.
Gore's success would definitely impact the GOP a fair bit and his appear toward the left and right would be seen as refreshing and maintain an odd hope in the American political system.
I do likely see 2008 being where Republicans take the White House back though they would be influenced by the years of Al Gore. They would definitely be more green-friendly and not as right-leaning. Granted, the Great Recession would throw things for a loop and if the GOP take in 2008, then while they would try putting the blame on the Dems for it, their likely mishandling of it would wreck them in 2012.