The short, accepted version is that it was the script and a simple lack of prep time. Like many movies, QoS was to be pumped out with very little time to prepare a script. Making things worse than usual, though, was the 2007 Writer's Strike, which would make continuing work on the screenplay impossible once the strike started.
The writing team behind Casino Royale, Neal Purvis and Roger Wade, returned to do a script which was then polished by Paul Haggis, as was the case with CR. Now, Purvis and Wade are controversial among Bond fandom. They worked on all the Brosnan Bonds, and tend to take the blame for those movies' shortcomings. A lot of fans, including the popular James Bonding podcast, will blame all the lame stuff on them, while crediting all the good stuff to additional writers like John Logan, Paul Haggis, and so on. This strikes me as unfair--a little like saying George Lucas never had any good ideas, and Star Wars and Indiana Jones's success was all due to collaborators.
I'm sure Purvis and Wade aren't Shakespeare or anything, but to hear them tell it, they've tried to write good scripts that are faithful to the Fleming novels, but have to abide by the wishes of various producers and directors. For instance, their initial script for Die Another Day would've been Craig-esque, focusing on Bond going after the traitor who had him tortured in a North Korean prison camp, and it was director Lee Tamahori who insisted on things like an invisible car, an ice palace, and so on, resulting in a movie that's half quasi-realistic spy thriller and half... whatever the hell the last half of Die Another Day is.
At any rate, that script was thrown out by director Marc Forster, with a new one written by Haggis, producer Michael G. Wilson, and Forster himself. This script was completed hours before the strike started, and under normal circumstances, it probably would've been worked on further. However, that script was then scraped to some extent: it would've featured a final confrontation in the Swiss Alps and Bond coming across Vesper's orphaned child (oddly, this subplot was scraped by the producers because they believed that Bond would choose to raise the child; wouldn't any reasonable audience accept Bond knowing he's an unsuitable father figure and seeing to it the kid is safe with Vesper's sister or whatever?). This left QoS pretty much without a script, as Daniel Craig admitted during an interview with Time Out London promoting Skyfall (which, like a depressing amount of sequel promotional campaigns, doubled as an apology for the last movie).
"On 'Quantum', we were fucked. We had the bare bones of a script and then there was a writers’ strike and there was nothing we could do. We couldn’t employ a writer to finish it. I say to myself, “Never again”, but who knows? There was me trying to rewrite scenes – and a writer I am not.’
You had to rewrite scenes yourself?
"Me and the director [Marc Forster] were the ones allowed to do it. The rules were that you couldn’t employ anyone as a writer, but the actor and director could work on scenes together. We were stuffed. We got away with it, but only just. It was never meant to be as much of a sequel as it was, but it ended up being a sequel, starting where the last one finished."
And QoS absolutely plays like Craig says. It seems like a collection of TV Tropes, subverted Bond clichés, and 'cool ideas' that could be good if well-executed, but instead are just trotted out in front of the audience to say their name like a Pokémon before the movie moves on to the next item on the checklist.
"Item 47. 'Critique how Bond movies will kill off the beta Bond Girl when James/the plot are done with her.' Oh, that's easy, we'll just literally have Dame Judi Dench say that's bad. NEXT SCENE!"
At some point in all this writing and rewriting, Quantum of Solace changed from being just another Bond movie, albeit one that would continue the QUANTUM plot thread (Bond's hunt for Vesper's boyfriend, as set up in CR, was always going to be followed up on), into Casino Royale Part 2. What I believe happened is this: initially, the movie that would become QoS would take a page from the literary follow-up to Casino Royale: Live And Let Die. It would start a few years after CR, with a Bond who still had unresolved issues from Vesper's death, but wasn't a complete trainwreck, and have him going on a mission for M that would end up bringing him back into conflict with QUANTUM, allowing him to get some closure on that.
However, CR was a massive success, with Bond and Vesper's relationship striking a similarly massive chord with audiences. The Bond producers have always tried to walk a tightrope of riding the zeitgeist while having something to do with what this guy Ian Fleming wrote. It's something Casino Royale director Martin Campbell was very good at, with both Goldeneye and CR. With QoS, though, everyone saw the praise for making Bond realistic and said 'screw being inspired by Jason Bourne, let's just make a Bourne movie that happens to star this British guy. And people liked seeing Bond's origin story--let's have this be more of his origin story! People liked Vesper? For the whole movie, he'll be depressed that Vesper is gone! JUST MAKE THE ENTIRE MOVIE ONE BIG CALLBACK TO CASINO ROYALE!'
And it worked. I would much rather be watching Casino Royale than Quantum of Solace.
I tend to lay the blame for this on Marc Forster, given that he made a similar hash of World War Z, a movie which doesn't have a story despite being based on a book with twenty different stories that all easily could've been movies. I'm sure Daniel Craig is hardly innocent, as he thought 'Quantum of Solace' was a good title for anything other than a fake play in a Wes Anderson movie, but Forster is a straight-up hack. Reading up on the film's production, he seems obsessed with the idea of centering the action sequences around the four elements.
This leads to a number of superfluous, and sometimes shockingly boring action scenes that are just in the movie because "AIR!" or "WATER!" or whatever, whereas action sequences that are more organic to the plot are given short-shrift, practically fast-forwarded through! Martin Campbell isn't a visionary, certainly--Green Lantern proves that--but he had very good instincts for how to advance the storytelling with action scenes. For instance, in the director's commentary for CR, he goes into how after Vesper's kidnapping, some drafts of the script featured a car chase with Bond chasing after Le Chiffre through a forest or some such. Seeing that this would be gratuitous, Campbell streamlined this to Bond starting his pursuit, but being run off the road--a moment right out of the novel. There, however, Le Chiffre used caltrops dropped from his car (the first Bond gadget, arguably enough), but Campbell changed it to them leaving Vesper in the road--neatly foreshadowing her betrayal of Bond as well as her unwillingness in the plot. It's good, good stuff.
Quantum of Solace, on the other hand, opens with... well, we'll get to that tomorrow.