Rapunzel is another name for Rampion. Bet you never knew that. The witch names her after the vegetable that was essentially being traded for her. The main characters in stories about "the maiden in the tower" were always named after a vegetable, Rapunzel is the name that stuck. Most of the elements apparant in Into the Woods come from the Grimm brother's version of the tale, which was derived from a version by Friedrich Schulz, which itself borrowed from Charlotte-Rose Caumont de La Force's version.
The story starts out with a man stealing vegetables from his neighbours garden to give to his wife, who claimed she would die without them. There is no mention of an older brother, or indeed any mention of the parents at all, once the witch (enchantress) takes Rapunzel away from them. The curse, and the beans, are in Into the Woods to link to the other stories.
In the story, Rapunzel ties her hair to a latch before lowering it, which you must admit is far more feasible, but less funny, than having her cries of discomfert. The Prince is drawn to Rapunzel at first by her singing, and later sees the witch's method for getting up there. Though many early versions mention an early consumation of their relationship, it is implied anyway by the twins that Rapunzel gives birth to after she is banished. Despite this the Grimm brothers still went to great length's to avoid it. At first Rapunzel gave herself up by asking why her clothes didn't fit, revealing she was pregnant. The Grimms changed this to a foolish slip of the tongue, in which she straightforwardly mentions the prince to the witch. The witch later refers to Rapunzel as the Prince's wife, as if to suggest the children weren't born in sin, yet there doesn't seem to be any possibility they could have actually been married.
The Rapunzel story, being the least prominant fairy tale in Into the Woods, didn't require nearly as much fleshing out. The insanity of Rapunzel is really the only major addition, axcept of course that the witch becomes a much more important character.