Executive Meddling: The original plan was to create extra material for each episode, then use it to edit each Story Arc into a movie for the DVD releases. But only the first arc was released like this before Sony switched to just plain episodes on DVD, leaving extra footage we might not ever see.
Doc Ock has two. Pre-breakdown is "D'awwwc Ock", post- is "DOCTOR OCTOPUS, LORD OF THE VAMPIRES" due to his rather vampire-esque attire in "Gangland".
Due to the various girls that Peter/Spidey attracts, fans have decided to rename the show "BITCHES LOVE SPIDER-MAN: THE ANIMATED SERIES".
I Knew It: In the first season, we are introduced to the Green Goblin. Most people by now know that Norman Osborn is the Green Goblin, and at first the show didn't do much to deny this, giving several indications that it was he. Then later that season, we were shown that it was in fact Harry Osborn. When Goblin made his return in the second season, the final episodes played up the idea that other people could be the Goblin, giving the audience some very serious misdirection. Of course, all of this was proven to be just subterfuge, as in the final episode it was revealed that the Goblin had, in fact, been Norman Osborn all along. Adding to the confusion, of course, was the fact that Harry Osborn was also the Goblin in the comics for a while.
A variant; Weisman has admitted that part of the reason Silvermane and Silver Sable are related here is because they both have "Silver" in their names.
Ricochet takes the name of another unrelated Marvel character for his superpowered identity.
The Other Darrin: After the first episode, Keith David is replaced with Kevin Michael Richardson in the role of Tombstone. In the Enforcers' debut, Clancy Brown did Ox's grunts, but when the Enforcers returned in season 2 and Ox actually speaks, he was voiced by Danny Trejo.
Screwed by the Lawyers: After Disney's purchase of Marvel Comics, Sony gave up the broadcasting rights of airing Spider-Man TV shows during negotiations to keep their film rights. Rather than drafting up a new contract to continue the production of Spectacular Spider-Man, Marvel and Disney decided to just go forward with a new show (Ultimate Spider-Man) instead.
Short Run in Peru: Canada aired the entire second season before the US even started on it. And then Disney put it on hiatus midway through…
And there were also rumors about Carnage (Cletus Kassidy was seen in season 2) as well as the Clone Saga (Miles Warren gained full control over Curt Connors' lab at the end of season 2).
Emily Osborn was to have been given a larger role and was to be voiced by Marina Sirtis, a.k.a. Demona.
Weisman's aspirations also included finishing the series with Peter's graduation from High School, then doing four straight to DVD movies, covering Peter's college years.
Christopher Yost was up for the executive producer job on Spectacular Spider-Man. They went with Weisman instead. 
Weisman said that had the show continued, he considered using Mr. Negative.
Weisman also planned to make Mary Jane to become Peter's main love interest at some point down the line. Which becomes interesting if we consider that he also decided to NOT kill off Gwen.
There's a small deleted scene from "Group Therapy" that explains how Shocker escaped from police custody and was free by "Probably Cause". However, it was cut for time and was meant to be included in the extended cut DVD release. Those plans were scrapped as well, so now it's shown by Weisman at conventions.