I would like a bit of history here, meaning I would like to know the difference in terms of creator, producers (or publishers of the comic if needed to explain the differences) and some explanation around the need of having two types of Spider-Man series (if it is for example a different producer, or Stan Lee didn't like one of the two or something like that).
I am particularly interested in the movie franchise where the two movie franchises almost overlapped each other.
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.
One of the main differences between the two series, is the fact that in The Amazing Spider-Man, it showed more of a personal storyline of Spider-Man and a strict detail of all of his actions in The Amazing Spider-Man movies. The original Spider-Man just showed sequence of events of a storyline, but in The Amazing Spider-Man, you see Spider-Man go through a real mission.
And another difference is that in The Amazing Spider-Man, the actors displayed more emotions and real feelings than the actors in the original Spider-Man movies.
I have watched all original Spider-Man movies and cartoon series but only movies of Amazing Spider-Man. So I can cite the following differences:
Original Spider-Man has web internally i.e. from with in his hands. Amazing Spider-Man had built a web system. But in cartoon of 90's he used cartridges there.
I found Amazing Spider-Man funny at times while there was hardly any funny moment in Original Spider-Man movies and even in cartoon series.
Mary Jane Watson was closest to Peter Parker in Cartoon and Movies in Original Spider-Man while in Amazing Spiderman movie, Gwen Stacy was.
The main difference is that one has an "internal webbing" (Spider-Man) while the other uses "web cartridges" (The Amazing Spider-Man)
At least this is what I remember from the cartoons.
