The first draft sets the story in the time of The Silmarillion, while in another, Bilbo mentions China and the Gobi Desert at some point, implying the story takes place in modern times.
Beorn was originally called "Medwed" (Russian for "bear"), Gandalf's name was originally to be "Bladorthin", Thorin's name was originally to be Gandalf, Smaug was to be "Pryftan", and Bilbo's working first name was "Bingo".
Plot-wise, the most drastic change was about Smaug, specifically that Bilbo killed him instead of Bard. In the earliest draft, Bilbo killed Smaug with a spear while the dragon was sleeping, while in the second and most developed one, the first time Bilbo infiltrated Smaug's lair he would stab him through the bare spot in his chest with Sting (which went so deep it vanished completely), and then ride a golden bowl on the stream of blood pouring out of Smaug's wound before triumphantly exiting the mountain. Tolkien decided to change it and invented the character of Bard to kill Smaug in a slightly more credible way.
After writing The Lord of the Rings, Tolkien wanted to go back and rewrite The Hobbit to include more Call Forwards and bridge the plots more smoothly. He actually started writing it, but he scrapped the project when his publishers told him that it lost the charm of the original.
The films were supposed to be simple duology, but it was changed to a trilogy. Contrary to popular belief, this idea didn't come from Peter Jackson, but from producer/production manager Zane Weiner, who was sort of a Hero of Another Story for keeping the production on track during its many troubles and helping by vetoing any outside meddling he was able to. Jackson nearly had an on-set nervous breakdown when it came time to shoot the Erebor scenes during the two-movie period and to plan out the Battle of Five Armies (which had been getting postponed up until the end of shooting because they couldn't find any locations in New Zealand that would've worked, and the battle turned out to be more complex than first thought during development). Weiner then requested the three-movie split to salvage the production and give Jackson the time he needed. You read that right, the three-movie split was designed to saveThe Hobbit. Still, whether this was a good decision or not is up for debate.
The other dwarves all had their own mini-arcs and character development that were shot, and both the main plotline and Gandalf's storyline were given equal screentime, but according to one of the actors, they were all cut out by the studio who demanded more emphasis on action and on Gandalf's story.
In the DVD Commentary Peter Jackson mentions that the Goblin Town sequence was initially imagined as a lot gorier, especially concerning the disembowelment of the Goblin King, but the PG-13 rating nulled those plans.
Smaug was also originally going to communicate telepathically with Bilbo (presumably through the Ring), since Peter Jackson was worried that if they moved his lips it would cause the scene to lose credibility. However after doing a few test shots of the scene, he and the other producers quickly realized that the telepathy route wasn't going to work, and changed their minds.
In the original script, Tauriel healed one of Bard's daughters (most likely Tilda) but when re-shoots happened it was changed to Kili, which coupled with the aforementioned Bolg switch suddenly explains Kili being hit with an arrow.
The Battle of the Five Armies
The movie was announced as There And Back Again. It was changed because Jackson felt that was more properly the title of the whole trilogy.
The concept art alone provides some really wild, potentially awesome scenarios, including a scene of Bilbo escaping some snake-like creatures by hanging onto a collapsing tower and running across to safety on the other side after it forms a bridge. The idea of an action setpiece on a tower was re-used for the Legolas vs. Bolg fight.
Unused concepts featured on the Extended Edition Blu-Ray's documentaries include a sequence of Gandalf having delirious visions through Dol Guldur's palantír. They would have featured Smaug leading Sauron's forces and Gandalf pursuing a weakened, retreating Sauron to the Sea of Rhun.
Jackson had plans to somehow get Aragorn to cameo in the film, fully intending to have Viggo Mortensen return to play him. However, as Aragorn was not even remotely a thought in the original book, this idea was shot down because Aragorn appearing in the movie didn't make sense timeline-wise. Instead, he gets offhandedly mentioned at the end of the third movie, setting up how Legolas and Aragorn know each other a bit.
According to the making-of book of the film, Radagast and Beorn were originally going to have much more screen time. Radagast featured a chase scene in his sled through Dol Guldur in order to evacuate Gandalf while the castle crumbled behind them; he then would have noticed Beorn being captured by some orcs, and he would have freed him before following with his mission, with Beorn escaping in bear form. Accordingly, those scenes were actually shot, being the reason why there is a Lego set of the film that shows Beorn being ambushed by two orcs in Dol Guldur, but sadly this footage wasn't included in the film's extended edition.
There was a scene where Bilbo motivated Bard and Gandalf and planted his acorn to remind them to keep fighting, even though it looks like they are going to lose. It was filmed and nearly made it into both the theatrical and extended cuts but was lost among the ten minutes of footage that didn't make it due to both timing and Executive Meddling. Unlike the previous one, this scene was released and can be viewed here.
There were also some additional Tauriel scenes, including one of her comforting a dying Laketowner and his family and another one◊ of her trying to shoot arrows at Smaug.
Thranduil's relationship with his wife was to be expanded upon and the explanation for the gems being so important to him as well.
A few awesome tracks were recorded for the soundtrack but aren't in the actual movie. For example, Dáin could have had this bagpipe-heavy theme tune.