The Hulk and Spider-Man are two characters with two types of expectations.
With a Hulk movie, people expect him to be a big, green brute that destroys things, more like a force of nature. The marketing for the 2003 film almost met that expectation, but the film didn't. It was directed by Ang Lee, who was known for his eye for action scenes (Crouching Tiger, Hidden Dragon), but also his penchant for character study (CT,HD; The Ice Storm; Sense and Sensibility), so it was virtually apparent that this wasn't going to be the expected Hulk movie; it was going to be a bit more ambitious in trying to explain why Banner becomes the Hulk, among other things. This is why Bruce Banner's origin was mostly grazed over as a freak accident in The Incredible Hulk, to get to the point of the Hulk smashing things.
Spider-Man is a nerdy-to-normal teenager who receives the abilities of a spider, and becomes a superhero. Balancing one's life between being a normal person and a superhero is a theme that runs through most superhero stories, so I think the expectation is there that conflicts will arise as Peter Parker and Spider-Man's lives collide. The "high school angst" angle doesn't bother me in that right for the new film series.
In addition, Spider-Man is a hero and his powers don't rely on his emotional state (unless the new series will follow the logic of Spider-Man 2), the Hulk is the opposite in regard to the latter and is not a hero (in the traditional sense), so there is no need to go as cerebral as Hulk did to understand the character. Not only that, but I believe that any studio with a superhero property is careful to study the history of adaptations to see what can work, and as this film is a "reboot" like TIH was, I think the Hulk/TIH case, and maybe moreso the Schumacher/Nolan Batman case, is being seen as an example of how to retell stories without retreading previous ground and how to meet and/or exceed expectations of how these characters should be adapted.
I'm more cautious of this film being a bit more like the production of Alien³, which involved a music video director (David Fincher) taking on the duty of helming a new installment of a well-known series into a grittier, more intimate direction that goes to the roots of what made the series popular (confined group + one xenomorph + no weapons). With a well-known series likely comes more studio involvement; this happened with Alien³, which may be why it garnered such a mixed reaction, and is a reason why Fincher isn't particularly proud of the film.
This new Spider-Man series has secured a director before production, unlike Alien³, but it still involves a music video director (Marc Webb) taking on the duty of helming a new installment of a well-known property into a grittier, more intimate direction that goes to the roots of what made the property popular (Spider-Man in high school). I'm thinking that Sony is also aware of the debacle that was the production and release of Alien³, even if it was so long ago, but too much studio involvement may still be something to look out for here, as well. (How coincidental is it that the writer for this new film worked with a post-Alien³ David Fincher on a well-received film?)