Ah, the old "organic vs. mechanical" webshooters debate.
Personally, I was open to the organic webshooters from a story-telling perspective. Raimi's films focused on the hard-luck Peter Parker, who is working to make end's meat, and who doesn't come from wealth. It would make sense for him to just have the organic webshooting powers instead of showing him build the webshooters and have the audience guessing how he was able to afford that equipment in the first place. The organic webshooters, in my opinion, aren't this monstrosity that fanboys have made it out to be, as I am more of a "big picture" kind of gal. The Raimi films contained quite a bit of dialogue that showed off Peter's scientific knowledge anyway (Spider-Man 2 being a prime example, with Peter seeing a flaw in Otto Octavius' project).
That said, I don't mind mechanical webshooters, either (because they are comic book accurate; it wouldn't make sense for Spidey fans to hate on the idea of using them). In the Amazing Spider-Man movies, there are some things they did right with them, and some things they did wrong. For example, in TASM2, the whole dilemma that Spidey has with the webshooters against Electro was actually a really cool idea ... But it was also really lame that Peter (a supposed super genius in science) needed Gwen to remind him of 8th grade science, and that he was relying on freakin' YouTube videos in attempt to figure the science stuff out. Also, in TASM1, I hate that Peter may or may not have stolen the equipment from Oscorp for the webshooters? If he bought them, then he technically didn't really make them himself ... And if he bought them legally, then who the hell sells that kind of stuff?! Wouldn't it be easy to find out he's Spider-Man in that case (not that TASM Peter would care about that stuff because of his enthusiasm for using label makers on cameras, or throwing footballs to dent poles)? Messy, messy writing.
All in all, I think the whole "organic vs. mechanical" debate is old. I like both, given that they are done right.