The Amazing Spider-Man is the name of the mainstream first spider-man series, set on in the main Marvel shared (with other comics such as the main X-Men, Incredible Hulk, Fantastic Four comics) universe that started in 1963 and is still going on with multiple releases in a month. I'm guessing the reason it's called Amazing is due to Spider-Man originally appearing in Amazing Fantasy before getting his own series.
That's where the new movie series got the adjective from. The reason they choose it is to distinguish themselves from the previous trilogy. I don't think it's meant to indicate that the movies aim to more closely follow the storyline of the main comic. For one thing the director of the new movies said
It's less based in Steve Ditko world and probably closer visually and more influenced by "Ultimate Spider-Man" but it is also very much a world of our own devising."
(Ultimate Spider-Man is an alternative comic line and continuity, modernized and meant to be more accessible to readers not familiar with 50 years of back story)
The reason for making a new Spider-Man movie universe so soon is financial probably because they made tons of money on the previous films, they also seem to be planning spin-off movies based on Spider-Man Villains), I've fairly sure I've heard rumours in the past that Sony loses the rights to Spider-Man if they don't make enough movies but I can't find any credible statement or rumor about that right now. The reason they didn't continue with the previous trilogy was due to disagreements with the producer Sam Raimi.