Exaggerated: Budou's past is a Trauma Conga Line of abuse, angst, suffering, and misery that would take years upon years of therapy to unravel. Her big fight with Ichigo causes tremendous damage and spills a great deal of blood. As soon as it ends they are the best of friends, never seen apart, and Budou is just as happy as Ichigo.
Downplayed: Budou is a child to loving, but poor, parents. Her childhood is troublesome instead of traumatic and allows her to become a magical girl using magic that's powered by greed. The nature of her power drives her to stealing and she is eventually stopped by Ichigo, whose magic is powered by friendship. The nature of her power leads her to be friends with Budou and help out at her place. They become good friends.
Queen Sangria has taken blood collected from one of Ichigo's battles with her and used her dark magic to construct a copy of Ichigo from it, who she names Budou. Although they share similar powers, the taint of evil has twisted her into something far more sinister. After being defeated by her twin, the dark magic's hold over her is exorcised, leaving her a more complete copy willing to join Ichigo's side.
All of the Magical Girls are from a Always Lawful Good Race. Budou was trained to be evil, but still instinctively knows she is not supposed to do what she is doing. When Ichigo defeats her, she offers her an opportunity to defect. She does.
Parodied: Budou is an obviously blood-crazed psychopath, but after being beaten by Ichigo, she is tied up in a scene reminescent of A Clockwork Orange and brainwashed with friendship speeches until she breaks and starts espousing the power of friendship on her own.
There is no direct counterpart to the heroic magical girl; Queen Sangria just uses a Monster of the Week tactic.
Enforced: "The boys in merchandising want another cute girl to put on the merch. How about giving our heroine a dark counterpart, so we can just use a slightly different palette?"
"Why couldn't I be a happy girl like you, Ichigo? Why does my fate have to be so harsh?!" "It doesn't have to be, Budou! I can help you if you let me be your friend!"
"Oh, come on! She wears black, has a Dark and Troubled Past, and is abused by her master! All you have to do is beat her up and rant about friendship, and she'd Heel-Face Turn in a heartbeat!"
Invoked: Queen Sangria intentionally sought out mentally scarred young girls that she could indoctrinate into her ranks and set against any would-be heroes. Likewise, Ichigo's goal in battle is not to 'defeat' but to 'befriend' because she wants to build allies while reducing Sangria's forces.
Exploited: Budou is Dangerously Genre Savvy enough to know that a Wide-Eyed Idealist like Ichigo would be easy to fool into thinking she's a DMG and so acts like one so she gain her trust so she'll be able to back stab her in the name of Queen Sangria.
After the first fight between Ichigo and Budou, Queen Sangria sees hesitation in Budou's eyes upon her return, and, refusing to take the chance that her servant could be turned against her, she obliterates her.
Deconstructed: Budou's emotional scars run so deep that they can't be healed just by losing a fight with her rival and listening to Ichigo's claims of acceptance. Ichigo can claim to understand all she wants, it doesn't matter to Budou. She continues to antagonize Ichigo until she either overpowers and defeats her or Ichigo is forced to take her down, sending the young heroine into a Heroic BSOD afterwards.
Ichigo freely admits she doesn't understand... She can't imagine what Budou's been through, and how she can still find strength when she's hurting so badly. But she knows Budou is strong, and that she doesn't deserve to be hurting the way she is. Budou sees that Ichigo really does have her best interests at heart, and Queen Sangria dosn't, and accepts Ichigo's friendship.
Played For Drama: The constant battles push each girl, Budou and Ichigo, into an emotional crisis. When they next confront each other, they decide to join forces - to defeat Ichigo's mentor as well as Budou's master.
Let's go back to Dark Magical Girl. Don't worry! As long as we're together, there's nothing we can't do!