California Doubling: The series was filmed and set in California, while the movie was filmed in Australia. The movie actually looks rather good for its budget, and some industry analysts have suggested it helped promote the use of Australia as a cheap filming location.
Deleted Scene: Actors at Conventions have commented on various scenes filmed that didn't make it into the movie, such as the Rangers doing some preliminary training between being given the ninja robes by Dulcea and traversing into the forest. In particular, Tommy's was apparently seen practicing some superhuman kicks similar to the corkscrew kick he used to finish off the last gargoyle.
The first fight scene in the construction site involved the Rangers fighting mutant, humanoid rats. However, it was universally agreed upon that the rats looked absolutely horrible, and they were replaced with the Oozemen. The rats later showed up in the TV series, in Season 2's "Return of the Green Ranger."
The Tengus were used in the third season as the Tengas.
Much later, one of the stone gargoyle-guardians the Ninjetti Rangers fight appeared in the background of an episode of Power Rangers S.P.D..
The Other Darrin: Zedd, Rita, Goldar, and Alpha are portrayed by different suit actors, though, thanks to their voice already being redubbed, they sound just the same as they did on the show (though Zedd sounds a bit deeper than his show incarnation). Their suits are slightly modified as well, to go with the Darker and Edgier feel. While Rita remains the same as she usually is, the "Z" on Zedd's forehead crest is notably lower down, he gains platinum pauldrons, and the gaps in his helmet's mouth are more closed up. His brain is also visibly pulsating. Goldar is inbetween the two: while he retains overall the same look, his face is more feline than monkey-like, and for some odd reason he doesn't carry his sword on him.
Goldar looking more feline-like makes sense when you consider that his original suit's face is supposed to be leonine, as his Zyuranger self is based on a griffon.
The Other Marty: Gabrielle Fitzpatrick was the original choice for Dulcea, but had to back out to remove a cyst. Mariska Hargitay (yes, that one) was brought in as her replacement. Filming ran longer than expected, along with some reported dissatisfaction with Hargitay in the role (ranging from her performance to how well she wore the costume) and Fitzpatrick returned. This actually resulted in discarding an extended sequence where Dulcea trains the Rangers in their proto-ninjetti powers (which are later seen in their fights), and they didn't have the time or budget to refilm with Fitzpatrick.
Adam being bummed out about being a frog was thrown in by Johnny Yong Bosch, who felt that it'd seem more real, and the fact that frogs aren't fast and agile, as he used to catch them in his backyard as a child. 
The directors of the movie, not having seen the actual series, pretty much did a quick "fast-fowarding" run of the series, ensuing things would go wrong.
When the Ranger suits were developed, the helmets were designed without visors or mouth pieces, intending on the heroes showing fear or worry. It wasn't until a little while later that they realized that they were meant to be a fearless force to be reckoned with and they remolded the helmets to include those missing pieces.
Gabrielle Fitzpatrick, the original and final choice for Dulcea, was replaced partway by Mariska Hargitay. However, after filming her scenes, which included a training sequence, they felt she wasn't the right one for the job and they rehired Gabrielle, dumping all of those scenes.
The filming for the final scene where the team is thanked turned into a disaster area when local radio shows caught wind of it and tried to turn it into a "Meet the Power Rangers" contest.
Originally, Ivan Ooze was meant to mutate a bunch of rats for the Rangers to fight. However, the rat costumes proved to be too low budget even for the actual series, leading to the creation of the Ooze Men. However, the suits were spared and used in "The Return of the Green Ranger" three-parter.
Delays in the series forced the cast to remain in Australia for filming, forcing Saban to make the aforementioned "The Return of the Green Ranger" episode.
The producers originally intended for the Rangers' masks to not have their visors and mouth pieces, feeling that it removed any sense of personality. They realized that their changes were counter-intuitive to the point of wearing a helmet because they were supposed to be an elite, faceless fighting force. They went overbudget and over schedule to refilm the necessary scenes. This actually resulted in the second season of the show overlapping with the film production (they were supposed to finish filming it when they returned home to America), the last few episodes were filmed in Australia.
Austin St. John (Jason), Walter Jones (Zack), and Thuy Trang (Trini) were originally slated to appear in the film, but due to their contract dispute halfway through Season 2, they were dropped.
After Ivan blew up his Tengu Warriors for failing, he created a colorful-feathered "Queen Tengu" out of their feathers.
Catherine Sutherland (Katherine Hillard, the second Pink Ranger) auditioned to play Dulcea. Evidently they liked her enough to bring her to America for a role. Sutherland has lived in America since.
Mariska Hargitay was cast to play Dulcea after the original actress, Gabrielle Fitzpatrick, had to back out due to a scheduled surgery. The producers didn't feel that Hargitay was working out and was replaced with Fitzpatrick after she recovered. Due to budget issues, she wasn't able to refilm an extended sequence where Dulcea trained the Rangers in their ninja outfits at her home. As such in the movie the Rangers seem to just sit and stare at the monument for a time before deciding to go.
Dulcea's backstory was in the script, but cut from the film. It's explained that she is the sole survivor of an alien race who created the power in the Ninjetti Temple. She and Zordon were also members of the same order that imprisoned Ivan Ooze in the first place.
Ooze originally created a team of rat-men to battle the Rangers at the construction yard. The director and producers felt they looked horrible and very low-budget (their goal was to create a high-budget looking movie) and such convinced 20th Century Fox to replace them with the more creepy looking Ooze-men. The rat-men costumes were later used in the "Return of the Green Ranger" three-parter on the TV series.
Instead of the gargoyles guarding the temple, the rangers would have fought another set of Tengu warriors - and their queen. The battle was supposed to happen on a long rope bridge on the way to the monolith. The set was built and put in front of a greenscreen but never used.
Dulcea would have had an annoying alien sidekick who picked on Kimberly. Around the time Gabrielle Fitzpatrick returned to play Dulcea, they decided to drop it.