Well, this was all quite deliberate: The producers had a set of rules stating that Spider-Man "couldn't throw punches, toss anyone through glass, put children in jeopardy, have anyone threatened by fire," or even say the word "kill" (so instead of getting murdered, Uncle Ben moves to Canada). That's also the reason why the grunge-era cops inexplicably carried around futuristic non-lethal blasters.
If you're gonna try and tase Electro, you're gonna need some better hardware, boys.
The most noticeable rule was probably the anti-punching one -- every fight scene turned into a predictable formula of Spider-Man dodging things thrown at him and then wrestling his opponents into submission before shooting his milky web goo and incapacitating them. Head writer/producer John Semper explained that the writers were told to always be careful that "when Spider-Man lands on the roof, he doesn't harm any pigeons." Either the censors didn't understand how animation works or the scripts were peppered with random acts of bird murder that we never got to see.
In contrast, please enjoy this compilation of Spider-Man punching people, getting shot with guns, and getting punched with guns in the 1960s cartoon: