These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Fridge Brilliance: Paul Soles was basically the only Spidey VA who actually tried to make his Spider Man voice sound considerably different from his Peter Parker voice which would help explain why nobody could recognize Spidey as Peter Parker when people such as Betty Brant hear Spider Man's voice.
Contrast the casting of a grown man as Spider-Man with complaints made during the 21st century about people like JoshKeaton and DrakeBell sounding too young to voice the web-slinger.
Similarly, compare the casting of a grown man as Spider-Man with complaints made during the 21st century about a 27-year-old and 28-year-old being cast as a live-action high school-age Peter Parker.
Memetic Mutation: Since the series became viewable on Netflix, people started watching the series and made a multitude of image macros based on it. Most of them point out the incredibly cheap and poorly done animation. Many can be found with a quick search for "60s Spider Man" on Google.
Nightmare Fuel: "Revolt in the Fifth Dimension," an episode intended for the final season, was never aired by ABC (which aired the season 1 episode "Sting of the Scorpion/Trick or Treachery" in its place) due to the network considering it to be too creepy for kids. They weren't too far off the mark; the episode itself is onehellof a trip.
Snark Bait: It's very hard to watch without laughing at, for some.
Uncanny Valley: Some of the villains' designs from the first season; close-ups of Doctor Octopus◊ and the Green Goblin◊ in particular (of course, this may have been the point). Every original villain from season two onward as well, some going straight into downright-Nightmare Fuel territory.
Values Dissonance: The show had other racial caricatures that, by today's standards, are incredibly racist. "One-Eyed Idol" has the savage Aborigine guide in employ of the Great White Hunter gunning for Spider-Man, "The Fantastic Fakir" features almost every Sim Sim Salabim trope in existence (complete with a villain whose entire theme is rope tricks), and the "Super Swami"... is actually racist in so many ways it's hard to nail down just what nationality he's supposed to be.
Sexist, too - women are usually either seen in subservient domestic positions or featured as crazed villains. Then again, Peter and Betty do seem to have a largely platonic working relationship.
What Do You Mean, It's for Kids?: The show was meant for a young audience, but a few times, death was actually mentioned, and someone actually was allowed to spout the word "Hell" at least once.
What Do You Mean, It Wasn't Made on Drugs?: With the second season's vast dependence on stock footage taken from a completely different science fiction cartoon, the plots became very strange, involving mole men, aliens trying to escape the Earth, and NYC's skyline constantly depicted as being filled with menacing purple clouds.