Enforced Method Acting: George Lucas intentionally put extra weight into the Darth Vader helmet so that when Hayden Christensen first walks around in the Vader suit, he almost stumbles, since Anakin/Vader's not used to the Vader suit yet.
Anakin was originally supposed to be watching the entirety of the duel between Palpatine and Mace Windu, but they realized that the implications were too subtle and somewhat confusing. This lead to reshoots that has the moment where Anakin and Padme "watch" each other across Coruscant and the duel had to be redone because Palpatine was using Anakin's lightsaber. As mentioned by a Lucasfilm executive, the original cut made it fairly clear Palpatine was playing up being helpless to get Anakin to side with him, while the reshot version made it seem that Mace Windu could have legitimately killed him.
A glimpse at the original backstory as seen in the novelization of Return of the Jedi (as well as strongly implied by Leia's lines in said film) shows us that who is now known as Padme was originally meant to survive Anakin's turn. Imagine the Drama that could have come out of that, as opposed to what many people viewed to be a cheap cop-out as well as a severe butchering of a character who fights a droid army alongside overwhelmed Jedi with her "losing the will to live".
Star Wars: Clone Wars ended up showing the actual kidnapping of Palpatine, which was present in the early drafts of this movie but then passed over to be depicted in that series. The opening battle was originally over 45 minutes in length even after they cut out that part, including one scene having Grievous execute Shaak-Ti in front of Anakin and Obi-Wan.
Earlier scripts had Boba Fett present and playing a significant role, Anakin turning into the black-clad cyborg we all love much earlier and not at the end, and a scene everyone would have loved: Jar-Jar contacting Padme... before being blown up.
There is a good amount of scenes that were not included in the final film, or even finished. These include two death scenes filmed for Shaak Tinote The Force Unleashed took advantage of this by turning her into a boss, Anakin and Obi-Wan treading through the sewers on Grievous' flagship, an entire subplot of Padme holding meetings with Mon Mothma, Bail Organa and other senators to plan the birth of the rebel alliance and a 30-second scene of Yoda arriving on Dagobah were excised.