One of the main differences between the two series, is the fact that in The Amazing SpidermanSpider-Man, it showed more of a personal storyline of SpidermanSpider-Man and a strict detail of all of his actions in The Amazing SpidermanSpider-Man movies. The original SpidermanSpider-Man just showed sequence of events of a storyline, but in The Amazing SpidermanSpider-Man, you see SpidermanSpider-Man go through a real mission.
And another difference is that in The Amazing SpidermanSpider-Man, the actors displayed more emotions and real feelings than the actors in the original SpidermanSpider-Man movies.