The Version You Know
There's never been a Disney version of this one, but you've probably heard it before. A miller has a beautiful daughter who he claims can spin straw into gold. A passing noble decides to call the miller on his shit and takes the girl and locks her in a tower and tells her to get spinning, presumably hoping to cause a collapse in the precious metals market.
Fortunately she's helped by a little gnome who shows up and offers to help in exchange for a small trinket. This goes on three nights, and by the third night the girl is promising the little man her first born child in return for his help. On the third morning, the king decides to marry this pretty girl who can produce gold out of dry grass.
They inevitably have a son, and the little gnome shows up demanding him. Being nothing if not fair, he'll give the girl three days to guess his name. If she can, she keeps the kid. She tries everything but comes up short, until a passing woodsman overhears the gnome bragging about how he's so clever no one will guess his name is Rumpelstiltskin. He immediately tells the queen, who springs it on Rumpelstiltskin, who's so pissed off he throws a tantrum and runs away, presumably to ply his poorly thought-out scam in another town.
What Got Changed
In the Grimm brothers' version, taken from the oral tradition, the little man is so pissed off he stamps the floor in his little hissy fit, and gets stuck. And then, like some insane version of a Will Ferrell skit, he pulls so hard to free himself that he tears himself in half. Now, if our names were Rumpelstiltskin and some dizzy miller's daughter had just told the whole damn room, we'd be pissed too, but we don't think we'd get dismemberment-angry.
Not to mention, in the really early versions of the tale, Rumpelstiltskin launches himself at the girl in a rage and gets stuck ... um ... in her lady parts. Seriously. The palace guards all have to come and pull him out, which must have made for some awkward looks afterwards.
Also, in a depressingly large number of versions, the child is killed anyway, either by Rumpelstiltskin himself, or the guards, or someone. They weren't big on happy endings in the Dark Ages. Plague will do that.