"Music In Your Soup" was to be sung by the dwarfs during dinner, and "You're Never Too Old to Be Young" was replaced by "The Silly Song". There were even more songs written for the film that never made it as far.
Sneezy originally had a verse during "The Silly Song". It was cut for time constraints.
Doing It for the Art: One of the main reasons Disney created this film, in addition to the potential cash flow, was his hope that it would raise animation to a higher art form. The film took four years to make and was not only a massive effort that required the studio to expand to over 1,000 artists, but there was no guarantee of financial success—the studio almost went bankrupt just from trying to get the film made, and it was only finished because Disney managed to get a large bank loan from showing a rough cut of the film. The short cartoons basically became a training ground for the monstrous amount of work and effects techniques needed to get the film done—the animators even had to take a few hours of mandatory, unpaid art classes after work each day in order to keep their jobs. Whole segments of the film had to be reanimated because the animators skills were so rapidly increasing. Disney was even doing color tests for scenes as late as four months into production. But it all paid off with what was (at the time) the highest grossing film of all time and one of the most influential movies ever made.
If one were to ignore the Guinness World Records' listing for highest-grossing film of all time when adjusted for inflation (which they awarded to Gone with the Wind, despite Gone with the Wind not making as much as Snow White and being released in a later year), Snow White is still the highest-grossing film of all time when adjusted for inflation, having earned $418,200,000 in its initial release note (according to IMDB, The Other Wiki, and several books, in contrast to claims made by Guinness World Records that it made less than half as much) which, with inflation factored in, equates to $4,500,938,448note (using http://www.usinflationcalculator.com/ as the calculator) as of 2016.
Dueling Movies: Technically, the final theatrical reissue of Snow White in 1993 did duel with Filmation's unofficial sequel to the film, Happily Ever After. This was after Filmation was hit with a C&D by Disney to stop that film's creation, and Filmation had already closed its doors, leaving the distributor to deal with the movie's release. Disney crushed the sequel and what was left of Filmation that year.
Follow the Leader: What with being the highest grossing film of all time upon release, it was only natural that Paramount Pictures would give Fleischer Studios the chance to make their own answer to Snow White, that being their animated adaptation of Gulliver's Travels. Even Walter Lantz, Paul Terry and Harman and Ising had plans to make their own features after the success of Snow White (none of which materialized, unfortunately). Snow White's success was also what convinced MGM to move forward with The Wizard of Oz, another fairy tale-esque/family musical, which unfortunately did not turn out well financially to start, but it also joined the Hollywood Masterpiece club in time.
Mean Character, Nice Actor: Lucille LaVerne as The Queen. If you look at one of her pictures, she looks nicer and friendlier and definitely less intimidating than the evil witch.
Money, Dear Boy: A big reason why Disney made this feature was that the Short Film product they made only brought in a set amount of revenue from their distributor regardless of how much audience interest in it. With features however, Disney's company could haul in much more if it proved a hit.
Old Shame: In recordings of Walt Disney commenting on the film, he said that after so many people seemed to dislike his films afterwards because none seemed to be as good as Snow White in their opinion, he grew to hate Snow White for a while. Though he eventually came back to loving his masterpiece.
Pigeonholed Voice Actor: Adriana Caselotti serves her sole feature film role as Snow White's uncredited voice actress, as part of her contract with Disney stated that she was never allowed to play another role again note there are exactly two other roles she played—as an uncredited voice speaking one line in The Wizard of Oz, and a very brief bit player part in It's a Wonderful Life or make public appearances to preserve the illusion of Snow White as a real character. Fortunately, she absolutely loved the role and openly embraced the reputation the film gave her later in life, which earned her the honor of a Disney Legend. Her own house was designed after the Dwarfs cottage and was filled with memorabilia from the film, and she even had an answering machine that responded to callers in-character!
The reason why the Prince had such little screentime was because the animators found him to be very difficult to animate.
Dopey was originally supposed to speak as well, but they couldn't find a suitable voice actor so he became The Voiceless.
Throw It In: An exaggerated example. When first animating the Dwarfs (save for Grumpy, who follows a little later) going outside to the washtub after Snow White tells them to wash up, Frank Thomas decided to give Dopey a small skip, or hitch step to his gait. Walt Disney liked it so much that he had several of Dopey's already-finished scenes reanimated to have their own hitch steps.
There were originally intended to be scenes with the Prince imprisoned by the Queen in a dungeon and making a daring escape, which was dropped because the animators were not yet experienced enough to handle the extra work of making a believable human character, what with Snow White and the Queen already taxing their skills. Some of these ideas were eventually used in Sleeping Beauty.
Some of the scenes with the Dwarfs were also cut, mostly to tighten up the story. Best known of these was the "Music in Your Soup" song, which was eventually shown on the Disneyland program in pencil test form. There's also a missing scene where the Dwarves build a bed for Snow White, which turned up in a children's book.
Sneezy had a line in "The Silly Song" which was recorded but cut due to time constraints.
All three of the Queen's assassination attempts (poison comb, bodice suffocation and the poison apple) were originally going to be included, but eventually they streamlined it to just the apple instead. The bodice was cut first, and the comb appears in the title card at the beginning. Because of this, a lot of people who grew up with the Disney version of Snow White think the Grimm's Fairy Tale version just had the apple attempt as the Queen's next plan to take out Snow White after the huntsman chickened out and tried to pass off a pig heart as Snow White's.
A proposed fate for the queen was shown in an Italian comic where she returned to the castle in hag form to ask the mirror who was the fairest in the land now. However, the Huntsman had already rallied a party to burn down the castle in order to put an end to her infernal majesty's reign. After receiving her answer from the mirror, she was burned alivewith the mirror just watching her suffer.
Alternate names and personalities were considered for the dwarfs, among them Wheezy, Awful and Jumpy. More information can be found here.
Dopey was originally going to have a voice, but he was made into the group's Silent Bob when an actor to provide the voice wasn't able to be found.
Deanna Durbin auditioned for the voice of Snow White, but was not chosen because Walt Disney felt her voice was too mature.
Dopey almost made an appearance in Fantasia as the Sorcerer's Apprentice.
At one time, Disney Toon Studios was exploring the idea of turning the Seven Dwarfs into a boy-friendly franchise much like Disney Fairies is for girls. An epic Lord of the Rings-inspired adventure that would explore how they came together as well as the origins of the Magic Mirror and Evil Queen was proposed, along with a Darker and Edgier video game prequel by Obsidian Entertainment about the Dwarfs' ancestors. This project ended up getting trimmed down in scope to a more comedic affair that would have revealed Dopey's muteness being the result of childhood trauma watching his mother die, but nobody had the nerve to present this idea to John Lasseter and he cancelled a further meddled version immediately.
A Snow White II was planned in 2002. It was about the evil Queen's sister Noriss learning from her magical crystal ball that the only way to make Snow White and her family leave the castle would be to kidnap her friends the dwarfs. She goes to the mine asking the little men to help her find her way home, then pulls out 7 bags to kidnap each one of them. Only Dopey escapes from her claws by hiding behind rocks. Snow White, the Prince, and their 13-year-old daughter Rose would've had to come to the rescue of the dwarfs with the help of Dopey.
"Snow White Returns" which was a proposed sequel short to Snow White and the Seven Dwarfs. It would have incorporated deleted material from the film such as the famous Soup-Eating Scene and a sequence of the Dwarfs building a bed for Snow White. The unmade short is detailed on the Diamond Edition release of Snow White.
In the original draft, the original character of the queen was supposed to be overweight and flamboyant. The prince was also supposed to be much more goofy and unattractive. Walt decided to change the queen to give her more of a serious personality and the prince followed soon after.