I was speaking with some of my students (I am a teacher) and even though the Raimi films came out just before they were born, all of them seem to prefer the Raimi films. Honestly, Sony should have worked harder to maintain a better relationship with Raimi ,Maguire and Dunst. That trio could do no wrong. Even though Spider-Man 3 was critically panned, the film grossed nearly $100 million more than its predecessor. Fans were very excited about the next installment.
Amazing Spider-Man did not meet the audience demand set up by the third Raimi film, but it had an amazing turn out for a reboot, especially compared to films like Star Trek and Batman Begins. Sadly, TASM2 seems to be losing traction. Most American movies, like most American businesses, generate most of their revenue overseas. The domestic box office is only marginally important. Pacific Rim for instance only had a lukewarm reception in North America. The film was a blockbuster overseas however, and that has led to a sequel. Even if TASM2 grosses $300 million domestically (highly improbable), it is not likely to make up sufficient revenue in the overseas market, which is why I placed my initial prediction around $550-650 million. This film will perform worse than the first film.
And as for domestic performance, Spider-Man may lose out to Captain America, which has led three weeks in a row as the number one box office film. The Avengers strategy has paid off, and people love seeing how the world building in Marvel films will lead to another exciting crossover. In a way, it is the old comicbook strategy come to life and working quite well in an all new medium.
On the positive side, Sony may wind up relinquishing their film rights to Spidey far sooner than anticipated, and that couldn't be a better development. I'd much prefer Spider-Man to be home with the rest of the Marvel universe.