Why the drastic overhaul of the Spider-Man costume for the sequel? Gwen helped redesign it.
Somewhat confirmed in the tie-in Infinite comic — Gwen is the one who suggests Peter get a new costume, and they begin working on the new design after the original costume is destroyed in battle, with Gwen suggesting the bigger eyes.
When we see the dying Norman Osborn, he is so sick that his skin is green and his fingernails are overlong. When Harry takes the spider venom, his skin turns green and his fingernails start to grow. It's entirely possible that Norman took the serum himself and suffered the consequences.
An interesting tidbit came up in the conversation between Norman and his son Harry. Harry mentions bitterly that his father gave him scotch for his 16th birthday. Norman himself probably downed a ton of scotch throughout his life because it gave some form of solace from the fearful knowledge that an incurable genetic disease was slowing taking his life, so he figures that his own son may as well start "partaking" early since he's going to meet the same fate unless a cure is found.
The location of Gwen's death being moved from a bridge to a clock tower was probably done to avoid being suspiciously similar to the climax of the first Sam RaimiSpider-Man movie, which adapted this scene partly. Non-comic fans would probably believe this film was copying that film's climax and this franchise has already been subject to many claims of being a cheap copy of Raimi's films.
Not to mention that Spider-Man saved not only MJ, but a whole busload of people in that version of the "Goblin dropping the hostage" scene. Changes were going to be made.
The clock tower also makes a better setting for giving a reason why Peter had no other options besides using his web to catch Gwen than the bridge would have, better than the original scene in the comics, even.
Catching Uncle Ben's killer was expected to be resolved in this movie, but the brilliance is it wouldn't have been good if it had. The reason is learning to let go of his petty vendetta and use his powers responsibly was a large part of Peter's character development in the first movie, and having him go back to searching for Ben's killer would have been a case of Aesop Amnesia.
Peter's phone playing the Spider-Man theme might seem to be Leaning on the Fourth Wall, at first; but, considering what a huge fanbase Spidey has In-Universe, it makes sense that somebody would have made the song as a tribute to him.
The clock hits 1:21 when Gwen hits the ground dead. Amazing Spider-Man #121 is the issue where Gwen died in the comics.
Electro playing his own music during the final battle actually does serve a purpose: It demonstrates that he now has complete control over his powers, enough that he feels safe wasting them in some fun ways while toying with a spider.
It's also a subtle reminder that the man that became Electro was (and still is) a superhero fan. He used to be one of Spider-Man's most devout fanboys, but now that he's written Spidey off as a Broken Pedestal, he's free to make up his own theme song on the fly as Electro.
When Peter talks to Harry as Spider-Man about not giving Harry his blood, he cites as a reason that their blood might not be compatible. It's entirely possible that they don't share the same blood type, which wouldn't help Harry if that's indeed the case.
After his incident with the electric eels, Max wakes up in a crematorium. That incident technically killed him. If he hadn't become electricity incarnate...
More squick than anything, but Max took clothes from the crematorium because his had turned to ash. Those clothes probably belonged to another body that had been cremated.
The Stacy boys have lost their father because of Spider-Man, and now Gwen has died because of him. Supervillains have been born from less.
If Electro is some kind of all-powerful electricity sponge, the consequences would have been far greater than a mere mundane blackout. He should have absorbed the electricity of people's nervous systems as well, killing everyone in the city. That would have been more realistic... and terrible.
The series has been cancelled to make way for the Marvel Cinematic Universe version. As this movie had a Bolivian Army Ending with Spider-Man in mid-battle with the Rhino, the audience can infer that Spidey died shortly after the credits began rolling.
Fortunately, this version of Spider-Man showed up in Spider-Man: No Way Home so we know he survived.
As pointed out by a comment on YouTube:
Gwen being as brilliant and smart as she is in these films, likely knows that even if the webbing catches her, the whiplash is going to kill her. So she takes one last look at Peter, closes her eyes, and waits.
