One of the complaints against the current Spider-Man comics was that Peter Parker was turned into Archie Andrews, a character incapable of change of growth or change. For these readers, there's no middle ground between a commitment to change approach, and having no growth at all. On the other extreme, some people wanted Marvel to go a lot further with their characters, truly adopting the Archie approach, asking for entirely self-contained storylines with no significant arc-to-arc continuity at all.
It had struck me that a single character running through a series, if it only engaged the reader, would bind that reader to that particular magazine. On the other hand, it seemed to me that the ordinary serial might be an impediment rather than a help to a magazine, since sooner or later, one missed one number and soon lost all interest. Clearly, the ideal compromise was a character which carried through, yet installments were complete unto themselves, so that the purchaser was always sure he could relish the whole contents of the magazine. I believe I was the first to realize this and The Strand Magazine was the first to put it into practice.
There are also some significant disadvantages to this approach. In his book Everything Bad is Good For You, author Stephen Johnson noted that stories in television and movies have become more intricate, relying on the audience understanding prior developments. Compare Arrested Development and Community to the Honeymooners, or the Lord of the Rings series to the Sean Connery James Bond films in terms of narrative complexity. This has also become true of comics, and the storytelling would seem infantile to anyone who has seen the Sopranos or Lost if the writers couldn’t reference prior continuity or relationships, or feature several ongoing (and sometimes) intersecting plots.
Stories also seem more significant if there’s the chance of major ramifications later. Readers won’t be concerned with any private subplot if there’s no possibility that Peter Parker will suffer a bit. However, the chance that a landlord will evict Peter Parker is more of a concern if the reader knows that the next issue can begin with a homeless Peter looking for a place to stay. A battle between Spider-Man and supervillain is more interesting if there’s the possibility that Peter’s going to be dealing with the injuries in the next issue.
This was the case with the pre-Paul Tobin Marvel Adventures Spider-Man. But Tobin quickly brought continuity to the all-ages title, along with longer arcs and subplots. Paul Tobin's been able to parlay success in all-ages comics more effectively than any writer since Dan Slott. Rich Johnston thinks it's possible that Tobin is the most widely-read comic book superhero writer today.
From what I've read of the Superman comics (much of Byrne's run, the first volume of the Showcase presents collection of the Weisinger era and a few Cary Bates issues) the Superman comics had a different structure, in that Clark Kent didn't change at all from issue to issue. Independent of the Superman plot, he almost always ended the story at the same place he began it. That's very different from Peter Parker, for whom stuff can change during the course of a story (he can be fired/ hired, piss off his aunt for the next year's worth of comics, dumped, meet a girl, etc.)
Spider-Man's life doesn't always change from issue to issue, but it sometimes did, going way back to Amazing Spider-Man #7. Usually it happened in little ways, with even minor events having consequences (IE- the principal calls Peter to his office to discuss a fight with Flash Thompson two issues earlier.) A key part of the series's appeal was to see what happens next to Peter Parker, and what the consequences for the actions would be.
It took the Superman comics a much longer time to adopt the idea of long-term subplots for Clark Kent, and they usually weren't as important for the book as Peter Parker's subplots were to Spider-Man. Let's compare a recent development in the Superman books (Clark Kent adopting a Kryptonian child in "Last Son") to a development in the Spider-Man books in the 1980s: Peter Parker's decision to drop out of college. Chris Kent tied into Superman's adventures, just as much as he did to Clark Kent, and when the "Last Son" arc came to an end, he went with it, leaving at the end of an arc.
To give another example, Gwen Stacy thinks she sees Peter Parker assault her elderly father in Amazing Spider-Man #60, causing a rift between the two. Her father remembers that Peter meant well in Amazing Spider-Man #64. She's able to find Peter and reconcile in Amazing Spider-Man #66. During this time, Spider-Man fights the Kingpin, Medusa, the Vulture, a bunch of prisoners and Mysterio.
That's the type of storytelling that quickly became a standard part of Amazing Spider-Man. And it took a lot longer to become a part of the Superman comics, which are often more traditional. It's still a part of the current Amazing Spider-Man, where there are long-term subplots with the supporting characters (such as J Jonah Jameson or Lily Hollister, the latter placing Peter in an awkward situation.)