Why this is a great movie
There are many reasons but a few stick out in my mind, all relating to the character and his presentation.
1. Lincoln is no goody two shoes, nor is he is an innocent naif. He comes off as homespun, genuine, unpretentious, modest, and idealistic. All those qualities are there. But he's also sharp, clever, canny, and unafraid to cheat or manipulate a little bit to achieve the higher good. Morally, he's a bit ambiguous and he's certainly no Capra hero (at least not a pre-George Bailey Capra hero) - unlike Mr. Smith, he always knows what he's up against and would rather play the game than stand on noble principle but be defeated. This is not to say he has no feelings, or that everything he does is calculated. At the same time, he's self-conscious, fully aware of life and death and knows that good does not automatically triumph in the world. He's a politician in the best sense of the word (and that sense is used rarely).
2. The movie's genius lies in its ability to present Lincoln the historical, mythical hero out of his usual context. It allows us to take everything we could take from a more traditional historical portrait, but without the obviousness of a Civil War story; furthermore, the power of that tramautic historical era remains all the more potent for sitting offscreen, hinted at by the brewing storm at the end, the occasional bit of music (the hymn in the beginning, The Battle Hymn of the Republic at the end, the snatches of Dixie throughout), and by occasional references to mankind's darker side (the lynch mob, the violence, etc). There's never a moment where we aren't aware of Lincoln's historical future, but yet the hints never seem too obvious; they're always at home in the film's context of 1837.
3. The subtle way in which Lincoln's character is established, that of an outsider who is crafty enough to use that outsider status to appeal to everyone. He's neither fully at home in the world of the poor (he can talk to them, but ultimately he's a well-dressed lawyer with prospects) nor the rich (he comes from a common background and is a bit too mystical and idealistic for their tastes) - yet he knows how to engage with both worlds and use them for his benefit. Once again, we see a Lincoln who is not entirely wholesome, nor a pompous fool. He knows how to play the game, but he has principles and he uses the game to achieve those principles.
Ford's direction is of course fantastic, as is the cinematography, the writing, and all the other elements. I think perhaps my favorite thing in the movie is Fonda's performance. Without it, the film wouldn't work for a second. But he achieves the ambiguity of Lincoln's nature, his ambivalence about acheiving prestige, and his good humor in the face of melancholy, always allowing us to see what Lincoln thinks and feels without ever completely letting us inside: a masterful performance, straddling the line between obviousness and obscurity, without ever coming down on either side, achieving a subtle and sure sense of character which embues the whole picture with life and gravity.