As I understand it, the only similarities between Master of the Universe (original title of Fifty Shades of Gray) and Twilight were the names of the characters (no vampires, sparkling or otherwise, etc.)
No, there are more similarities. For instance, Christian's adoptive family is a blatant rip-off of the Cullens: all very beautiful, very polite people. His sister is exactly like Alice (minus the whole "I can see the future" thing), his brother is clearly Emmet, who Anastasia's best friend starts dating. Also, like Rosalie, Kate (Anastasia's aforementioned best friend) opposes their relationship.
Then, there's Anastasia's and Christian's relationship. She is frequently dazzled and rendered speechless by him, not to mention she sees him as an Adonis (yes, this is in the book). She is so completely entranced by him that no one else matters (à la Bella Swan) and it takes someone stupidly handsome to make her want to date and have sex. Christian, like Edward, is also frequently borderline stalking her.
Also, let's not talk about the whole misogynist theme going on in Twilight - but suffice to say, it's present in FSoG as well.
So, the resemblances between Twilight and Fifty Shades of Grey go beyond character names that ended up being changed for publication. The main relationship, the relationship between Anastasia's and Christian's family, the characters themselves... they obvious belong to a fanfic, not to an original novel.