I think that doppelgangers are not only identical in appearance but identical in persona and character. It’s just the environments in which they are raised that make them different.
That being said, Katherine and Elena are different is because of how they grew up.
Don’t you remember in 2x19, the flashbacks? Katherine seemed SO MUCH LIKE ELENA. Sweet and innocent….and naive..and we still don’t know what Elena was like pre-accident. She said herself that she was more fun and reckless…more Katherine-like.
And now Elena is slowly becoming more Katherine-like in her actions as well.
And JP’s interview says: “So who is [Elena] really at her core? She’s clearly not the girl-next-door do-gooder, she never has been.”
So the point I’m really trying to make here is this: Elena and Katherine have the SAME NATURE. It’s just their nurture that made them different.
But are they really that different? Is Elena eventually going to end up like Katherine?
I think for some duration of TVD she will, yes. It’s one of her biggest fears - turning into her - but I think it will happen. There’s been anvil sized hints all throughout season 3 about her humanity and what not. I honestly think that she’s going to turn into a vamp in the season finale and next season will shut off her emotions - i.e. become Katherine 2.0.