When Toy Story was in production, the head of Disney was Jeffrey Katzenberg, who did not care for the rather soft tones of Woody and Buzz's relationship. Katzenberg wanted a new script, something "edgier," with more "insult humor." In the second draft of the film, the one production started with, Woody was considerably more of a selfish jerk, as opposed to nervous and insecure, far more sarcastic, and worse with the insults... which, in a film featuring Don Rickles as Mr. Potato Head, is saying something.
When Roy Disney saw the story reel, he called it "the worst thing I've ever seen," and the entire project came within a hair of being simply cancelled. Instead, Pixar was told to go back to their original concept, while reusing as much finished footage as they could, and the film as we know it took shape and was released. It was the only Pixar project that Katzenberg had anything to do with, as he left Disney rather suddenly before the film's release.
Due to the appearance of Pizza Planet, a pizza restaurant, in the first film, there are now at least two Pizza Planet restaurants in the real world.
In the first draft of the script, Andy, the boy whose toys Buzz and Woody are, had both a mother and a father. This was changed to his being raised by his single mom (Laurie Metcalf)... because the film's budget wouldn't allow for the extra voice actor and the model development for another human character.
Barbie and Ken were slated to appear in an early draft of the script, but Mattel nixed the idea; they didn't like the idea of Barbie having a PERSONALITY, since Barbie is supposed to reflect whatever any little girl who owns one wants her to BE, and they were afraid that if the movie painted her in an unflattering... or merely DIFFERENT... light, it would hurt sales. This is why Bo Peep was Woody's love interest in the first film. Mattel changed their mind after the wild success of the franchise, which is why Barbie appears later... but will not likely appear in Part Four, as Sony now has Barbie's movie rights.
Interestingly enough, while a number of Buzz Lightyear toys have been made, following the success of the franchise? There has never been a Buzz Lightyear figure that included every single feature that Buzz had in the movie (apparently, there's one with pop out wings, and another with Karate Chop action, but not one with both, as the mechanisms would have to occupy the same space).
EDIT! This link leads to an eight minute chunk of footage on YouTube that includes one of the cut scenes of Woody's "edgy" character. John Lasseter cringes at its badness, but admits it's necessary to show it for history's sake. Woody coulda been a serious (broccoli.)