Tom Bombadil among others played a significant part in the books but were left out in the movies. Was this just to save time?
Tom Bombadil as a creation precedes much of the LOTR mythology. He appears first in a 1934 poem "The Adventures of Tom Bombadil", where he is a "'merry fellow' living in a small valley close to the Withywindle river, where he wanders and explores nature at his leisure." (https://en.wikipedia.org/wiki/Tom_Bombadil).
Bombadil does not really fit into the LOTR mythology, at least, there is very much discussion where to place him exactly: http://tolkien.slimy.com/essays/Bombadil.html So he very often gets left out in adaptions, as he doesn't really fit in all that well.
More to the point, the director Peter Jackson is quoted as saying that, while he first had plans to use Tom Bombadil, he left him out because he didn't advance the action: "the Bombadil sequence has so little to do with Sauron or the Ring, it is difficult to justify the screen time. It simply doesn't give us any vital new information." http://www.theonering.com/complete-list-of-film-changes/the-fellowship-of-the-ring
He didn't play a signficant part in the books. In terms of plot pacing, his role was to provide a little bit of down-time between the tense times of Old Forest, and the Barrow-downs .
Tolkien's style of plot pacing was that after the characters go through some trial, there should be a relaxation period before the next trial. (This is very common in film and print).
So in the book, we had:
- Crickhollow (relaxed)
- Old Man Willow (tense)
- Tom Bombadil (relaxed)
- Barrow-wight (tense)
- Tom rescues Frodo, and segue to Bree (relaxed)
The first film actually uses similar pacing but skips one cycle:
- Meeting Pippin and Merry (relaxed)
- Black Rider chase scene (tense)
- Reaching the ferry, segue to Bree (relaxed)
To keep with this pacing pattern, if the film included Tom Bombadil after the Black Rider Chase, it would have also meant including another scary scene such as the Barrow-downs. From this point it is speculation; but the scriptwriters must have felt that adding these two scenes would either add too much running time, or take the viewer too far away from the main narrative.
Screenwriter Phillipa Boyens explains in the extended edition documentary that the central message they wanted to emphasise with respect to the one ring is that no-one (not Gandalf, Aragorn, Elrond - not even Frodo at the end) can resist the lure of the ring. Then, along comes Bombadil and it has absolutely no effect on him. Including the Bombadil sequence undermines that message about the ring.
I will add my bit.
Tom Bombadil adds excitement and appears significant because the book through this point has been dreadfully boring. That may be too harsh, but the first half of The Fellowship of the Ring moves along at a glacial pace. The two almost-encounters with the Nazgul are really the only action and are very brief. Old Man Willow is the first real extended action in which the hobbits fare poorly and only Tom showing up and rescuing them saves the day. The scene in the barrow downs is also quite exciting and the hobbits once again fare very poorly and only Tom showing up and rescuing them saves the day.
So the first two episodes of excitement and suspense are both entangled with this great and mysterious figure, and comes to mind prominently in the minds of readers.
In the narrative of the overall story, however, Tom really isn't that significant in moving the story along.