Tolkien himself said that Bombadil had no real place in the story. It wasn't a loss.
This is a slightly modified version of a post I made on another forum on this topic.
While I think that Bombadil is a character who is not really missed from the films, and who probably would not work had they decided to keep him; at least in the film that was made, to say he had no place in the book is an error. To not like him on a personal level is one thing, but to disregard his importance is another. Indeed Tolkien only kept in that which had meaning to his story as he told it.
I like Bombadil and what he represents. He more than any character (even Gandalf) is Sauron's antithesis. Both have immense innate power, but where Sauron wants total domination over all things: Bombadil wants domination over nothing. It is for that reason the ring has no power over him, and precious little over Hobbits (it also explains why Faramir wasn't tempted, at least in the book, but that is a different debate). But it is precisely because of that he is of no use to the enemies of Sauron: he has no interest in anything outside his little corner of the world.
I'll defer to someone who knew infinitely more about these things than I.
quote:
J R R Tolkien
From a letter dated 25 April 1954
...Tom Bombadil is not an important person - to the narrative. I suppose he has some importance as a 'comment'. I mean I do not really write like that: he is just an invention (who first appeared in the Oxford Magazine about 1933) and he represents something that I feel important, though I would not be prepared to analyze the feeling precisely. I would not, however, have left him in if he did not have some kind of function. I might put it this way. The story is cast in terms of a good side, and a bad, beauty against ruthless ugliness, tyranny against kingship, moderated freedom with consent against compulsion that has long lost any object save mere power, and so on; but both sides in some degree, conservative or destructive, want a measure of control. But if you have, as it were taken 'a vow of poverty', renounced control, and take your delight in things for themselves without reference to yourself, watching, observing, and to some extent knowing, then the rights and wrongs of power and control might become utterly meaningless to you. It is a natural pacifist view, which always arises in the mind when there is a war. But the view of Rivendell seems to be that it is an excellent thing to have represented, but there are in fact things with which it cannot cope; and upon which its existence nonetheless depends. Ultimately only the victory of the West will allow Bombadil to continue or even survive.Nothing would be left for him in the world of Sauron...
While he isn't missed from the film, I do not believe the film is better because of it. He doesn't do much to progress the story, but the book is more than a series of plot points and set pieces. As much as I like the films (and I do despite there being certain choices made I fundamentally disagree with) they are in no way near matching the scope of the books. That is not a slight on the films but a comment on the sheer scale of the world Tolkien created.
I don't think any film-maker could possibly match it, although Peter Jackson and co. came closer than I ever thought possible.