Of course the story got changed,the origin from Amazing Fantasy 15 have nothing to do with organic webs.
Plus i always liked several aspects of the web shooters of Spider-Man,as that being something he invented right in the first stories thus showing the genious Peter Parker was in the first stories or Spider-Man runing out of web cartridges and so on.
The organing weebing was always a extra power that i found to be silly.