According to Original Cast Precedent, the Narrator and the Mysterious Man are played by the same actor. Same goes for Cinderella's Prince and the Wolf, as well as Cinderella's Mother and the Giantess and Granny. Usually played in an And You Were There fashion, at least for the first two actors.
In the Broadway revival, Cinderella's Mother was played by a recording of Cinderella's own actress.
In the Central Park production, the Narrator was changed to a young boy so instead the Mysterious Man doubled as Cinderella's Father (played by Chip Zien, the original Baker, in a nice Remake Cameo). And they got Glenn Close to lend her (prerecorded) voice as the Giantess.
Cut Song: "Giants in the Sky", "On the Steps of the Palace", and "Last Midnight" were all replacements for less appropriate songs written earlier. There was also a song called "The Plan" that was cut. However, the initial versions of those songs are included on the original soundtrack; the prototype of "Giants In The Sky" is notable for being so fast-paced that it's nearly unsingable.
Genius Bonus: In the original fairy tale, Rapunzel is named after the plant her mother craves while pregnant with her. This plant is also sometimes referred to as "rampion," which the Witch is the most upset about the Baker's father stealing from her, making it a stealth reference to the original story.
Knowing this also explains why Cinderella's Prince makes fun of Rapunzel's name: not many people share a name with a vegetable.
Both Chris Pine and James Corden have admitted that the entire reason they pursued parts in the film (as Cinderella's Prince and the Baker, respectively) is because they heard that Meryl Streep was playing The Witch From Next Door.
Streep herself had consistently refused to portray witches in film for a long time, but made an exception for this film because she wanted to be in a Stephen Sondheim musical.
"I Guess This Is Goodbye", "Maybe They're Magic", "Our Little World", "First Midnight", "Second Midnight", "Ever After", "So Happy", "Agony (Reprise)", "No More". Interestingly enough, several cut songs appear in the scenes as background music where the songs were. Listen to the music as the Baker mourns his wife's passing and you'll hear "No More". "Ever After" is the march played as the first "act" ends at Cinderella's wedding, and it transitions into a snippet of "So Happy." Technically, "Cinderella At The Grave" is not cut, but only the mother's part remains (although the "shiver and quiver little tree" melody is heard as Cinderella's dress and slippers appear on her. Its short reprise (where the ghost and birds warn the prince about the slipper) is also cut; the Steward simply notices the blood when assisting Florinda, and Lucinda exposes herself by being completely unable to walk. The melody of "this is the proper bride for you" does return when the prince puts the slipper on Cinderella's foot.
Also, the film originally had two new songs, "Rainbows" for the Baker and his wife and "She'll Be Back" for the Witch. Both were cut out of the final film.
A few of the songs were also rewritten. "On The Steps of the Palace" is now primarily in the first person instead of the second, while "Your Fault" is noticeably slower, most likely because the actors couldn't keep up with the pace of the original.
A more likely explanation for the adjusted tempo of "Your Fault" (and other tongue-twister lyrics, like the "Witch's Rap") is that filmmakers thought the wider movie audience would have more difficulty following such fast-paced songs than a theater audience did.
Hide Your Pregnancy: Emily Blunt was around 4-6 months pregnant throughout the shoot, so there were many scenes of her lower section being hidden. Amazingly, Blunt's face doesn't look as full as most pregnant women's faces are, so it was hard to tell if she was at all.
Emma Stone was Rob Marshall's first choice for the role of Cinderella but she turned it down, feeling that she didn't have the vocal range. She joked that she'd be better suited to playing Jack instead.
An early draft of the script contains several elements that were altered (or never used) in the final film, such as...
The depiction of the Wolf as a sexy, hairy chested man. He also briefly transformed into a real wolf right before he howled during "Hello Little Girl".
The fate of the Witch. Rather than being swallowed up by a sinkhole, she is pulled into the ground by the arms of her mother, much like the outdoor productions.
The fate of Jack's mother. In the early draft, she is bludgeoned by the Steward as in the stage show, but she manages to get back onto her feet and temporarily help the Baker and his wife find Jack. She eventually succumbs to her head injury, and the Baker comes across her dead body in the woods. This explains how he had found out about her death prior to telling Jack during "No One is Alone".
The original finale. In the draft, all of the characters appeared at the end to sing "Children Will Listen", much like the original show. Afterwards, everyone would "disappear" (sans the Baker, Cinderella, Jack, and Little Red), and the rest of the song (a reprise of the title song) would be sung as a voice over. In the final film, the entirety of "Children Will Listen" (after the Baker's Wife's solo) is sung by the Witch and the chorus off screen, and the "Into the Woods" reprise is played over the end credits.
The scene where the Baker cuts open the Wolf and saves Little Red and Granny. The scene was originally portrayed as a "shadow play"; all of the action would be seen as a silhouette over the bed drapes (the scene also contained the full conversation between Granny, Little Red, and the Baker from the show). The final film cuts away before the Baker brings down his knife on the Wolf, and the entire conversation is greatly shortened.