Deliberate mistake: Bowser has several teeth that appear and disappear throughout the movie. The best example of this is when Mario is trying to grab the Power Star. We see Bowser snarl at Mario before he picks him up. There is a gap between his two front teeth. However, when the scene cuts from Mario's horrified face to Bowser's face, there are now extra teeth inside that gap.
Continuity mistake: When Mario first consumes the Blue Mushroom, he emits a light that causes Donkey Kong to shield his eyes. However, later in the movie, when Peach forces Bowser to consume the same mushroom, no light is emitted. (00:49:15 - 01:21:22)
Continuity mistake: Peach's crown suddenly appears on her head a few cuts after taking her helmet off.
Continuity mistake: During the kart ride to Cranky Kong's castle, a wide shot of the kingdom is shown, showing only two ramps on either side of the entrance. However, moments later, there is now a third ramp in the middle, leading to the entrance. (00:42:42)
Answer: "Princess Toadstool" was used in the original English-language manual, but she was Peach in the original Japanese, and that name continued in later versions, being combined with Toadstool. Bowser was originally called "Kuppa", but presumably the English-language version resonated more with people, plus removed any confusion of "Kuppa, King of the Koopas", so the English version stuck.