Accidental Aesop: Every time the show hands us a lesson about great power and responsibility and the right thing in conjunction with Peter's keeping his identity secret, it ends up telling us that secret identities are a stupid idea and nothing good can ever come of them. Which is more or less true, if largely inapplicable toour daily lives.
J. Jonah Jameson. Daran Norris nails it, making Peter's ill-tempered boss hilarious.
Gwen's portrayal is one of the major reasons for her surprising resurgence in popularity. Her design (especially since it becomes closer to her comic counterpart's own) and voice are very, very adorable and she's got a Cool Big Sis dynamic with MJ. Though her film incarnation is particularly different, she's the other half of the reasoning. When it's not her, fans tend to think of this version. The Spider-Gwen comics seems to split the difference between the two.
The entire Symbiote arc. Venom—or more accurately, the symbiote itself—acts as a spurned ex/stalker towards Peter Parker. Eddie Brock doesn't seem to have much issue with this either, judging by all of those unsettling leers aimed at the web-head. Of course that could just be the symbiote's influence, but it's certainly there regardless. The symbiote doesn't seem to have any problem going on about how much it just wants to be with Spider-Man. Spidey even refers to himself (albeit somewhat ironically) as its "first love" at the end of "Nature vs Nurture". Venom seems to hate Spider-Man just as much as it wants him back/to itself.
Norman Osborn's obsession with Peter Parker already borders into Ho Yay territory (not that Peter helps matters, Normie-kins indeed...), but Spider-man and the Green Goblin?
The first five episodes are entertaining; however, episode six abandoned the usual "two fight" format, going for one big fight with The Rhino instead, and then introduced Tombstone.
Season two also introduced much more complicated, long-running plot threads and twists, and generally amped up the quality of the fight scenes, which were already arguably amongst the best of Western Animation as is.
Ham and Cheese: Mysterio, who milks his sorcerer shtick for all its worth. As Tinkerer mutters, "Actors"
Hype Aversion: Unfortunately the fandom has caused this, due to their boasting that this is absolutely the best Spider-Man adaptation ever and not accepting any other opinion, as well as their consistent (and often times unnecessary) bashing of Ultimate Spider-Man and its fans have scared off a lot of people who missed this show, and caused them to not want to watch it, either due to disliking said fandom's attitude or feeling it won't stand up to the sheer amount of hype and praise surrounding it.
Magnificent Bastard: Tombstone, Dr. Octopus, and Norman Osborn. Their schemes run rings around every other character in the show. Ock is to the point that his in-universe title is the Master Planner, and Spidey spends a fair few episodes just trying to find out who the heck the Master Planner even is!
Norman framing Harry solidifies him as an amoral bastard, but injuring Harry's leg to do it demonstrates a casual cruelty and cowardice that erases any hint of redemption. Though manipulating Mark Allan was also up there.
Jonah's behaviour in "Identity Crisis" is pretty despicable, taking the utmost joy in the possibility that his teenaged photographer is Spider-Man, fully knowing that people would want to KILL HIM and anyone he knows if his identity is revealed.
Indeed, Venom crosses it by revealing Spider-Man's identity in the first place knowing that he and his loved ones will be threatened.
Narm: Eddie Brock, post-Venom, in his moments without the symbiote.
Eddie: It only loves me for the hate!
Older Than They Think: Many who are aware of the symbiote since the 90's animated series probably didn't know that in its original comic book appearance, it was able to take over Peter's body in his sleep.
Twas the night before Halloween, and all through Manhattan,
Not a creature was stirring, not even Green Gob...lin.
Paranoia Fuel: Eddie Brock uses this to make Peter reveal the location of the symbiote, allowing him to become Venom once more.
Periphery Demographic: Many of the show's fans appear to be adults. This is most likely because Greg Weisman was working on it, who has many fans that were children when Gargoyles came out and by the time Spectacular was created they have grown up. The top-notch writing and fight scenes don't hurt either.
Rewatch Bonus: After Peter gets the symbiote, it starts speaking to him (in his head). You won't realize this until Pete does.
The Scrappy: Good luck trying to find someone who's watched the show and doesn't find Sally annoying.
They Wasted a Perfectly Good Plot: When John Jameson has difficulty in space and all of Earth is transfixed by it, Randy Robertson is particularly worried and says that John is "like an older brother to him". Given that their fathers (J. Jonah Jameson and Robbie Robertson) work closely together, this relationship makes perfect sense. However, nothing more of it is ever seen or spoken of, even after John is driven insane. It seems like in juggling the Loads and Loads of Characters in this show, the ball was dropped on Randy.
Peter Parker, during his relationship with Liz Allan. He doesn't even attempt to hide that he has feelings for Gwen (or doesn't even realize he shouldn't be making it that obvious) even when right in front of Liz.
Not to mention he and Gwen deciding to break up with Liz and Harry at the end of Season 2. This is coming right after Peter spent all season not showing up to his and Liz's dates (admittedly for crime-fighting), showed favoritism to Gwen over her on Valentine's Day, and Mark Allan's arrest for supervillain activities. Not to mention the fact that Harry was kidnapped by the Green Goblin and a struggling addict who had his secret outed by Flash a number of days prior.
In fact, they decide to get together just minutes after Harry tells them that he's spent the last several days kidnapped!
Also, his attitude towards Dr. Connors; when Connors became the Lizard, he took a picture of himself fighting Lizard, making it look like he (as Peter) had been hiding and choosing to do his photograph job over helping Connors. Now, this could be forgivable as a one-time mistake, but he does it again in Persona, when he takes pictures of his fight with Black Cat over the symbiote, making it look again like he couldn't even bother to call the police. Even worse, Brock actually told him he couldn't take pictures right before precisely because of the trust issue.
WTH, Casting Agency?: Tinkerer's voice makes him sound like a child, when he's supposed to be an old man. It's obvious that the voice was misused, since years later, Adcox would indeed use the same voice when cast as a child. However, that's Adcox's actual voice, and he's about as old as Tinkerer.
The Woobie: True to the comics, the sheer amount of crap poor Spidey gets put through over the course of this show, particularly from the first and second seasons, is unbelievable.
Jerkass Woobie: Just like in the comics, Harry is this. He may have ditched his friends when he was In with the In Crowd and manipulated Gwen into staying with him, but given how his home life is and how he's treated in general, you can't blame him.