"Snow White and the Seven Dwarfs" is an ancient story. It's one of those stories that's so old that it was passed down orally for a long time before it was ever written down, and its precise origins are now unknown.
The Brothers Grimm, however, were kind enough to finally write it down (under the title "Schneewittchen" or "Little Snow White") in 1812. This was the version that became the definitive story and is the one on which Disney's Snow White and the Seven Dwarfs is officially based (though it's likely the 1912 stage adaptation was an influence).
While the broad story and biggest plot points remain largely the same, the Grimm boys' version of the tale does differ from Disney's version in several interesting ways:
1) We briefly meet Snow's mother at the start of the story. She pricks her finger and is fascinated by a drop of blood in the snow next to her black window frame. She wishes for a daughter with skin as white as snow, lips as red as blood, and hair as black as the window frame. It's not really vital to the story, but it might have been nice to include in the animated film - seeing the lost mother, and offering a little explanation for the title character's odd name.
2) Snow White does take refuge in the dwarf's cabin and befriends the group, but not only don't the little guys have individual names or personalities, they really barely register as part of the story at all. Disney's seven dwarfs are a huge improvement on the source material.
3) The queen makes three attempts to kill Snow White in three different peddler disguises (all costume and makeup, no magic transformation). The first time she binds the girl's corset so tightly she can't breath and leaves her for dead (the dwarfs arrive and unbind her). The second time she combs her hair with a poisoned brush and again leaves her for dead (the dwarfs come home, remove the comb, and she's fine). Finally, she does the old apple trick and that one lasts a lot longer. Now it seems surprising at first that Disney took out the first two attempts. After all, when taking a short story and stretching it to feature-length, you usually need MORE drama, confrontation, and plot points, not less! Still, Snow falling for mild variations on the same trick three time and the queen leaving, assuming the job is finished three times kind of serves to make them both look pretty stupid, and I think the Disney version is stronger for the alteration (and, as I said in her character profile, I love the addition of the magical transformation into the old hag).
4) The prince does not meet Snow White before her journey into the woods or her poisoning by apple. He sees her for the first time in her deathless sleep in the glass coffin and becomes obsessed with her there. He first tries to buy her from the dwarfs, and then convinces them to give her to him because of his love for her. It's... pretty creepy.
5) It's not a kiss that wakes her, but one of the prince's servants stumbling while carrying the coffin. This dislodges a bit of apple still in her throat and revives her. Had any of the dwarfs known the Heimlich maneuver, she would've been fine much earlier. True love's kiss, again, is a big improvement.
6) The queen gets a very different comeuppance. She's captured and forced to dance wearing red-hot iron shoes until she drops dead. It's wonderfully dark and weird, but I think I understand why that change was made.
For a lot more on the origins of "Snow White" and lots of other info regarding the story, check out this scholarly Snow White page.
Illustrations in this post are by early Grimm illustrator Walter Crane.