Although the Asuka of the Neon Genesis Evangelion tv series and the Asuka of the Rebuild of Evangelion films are essentially the same character, there are marked differences that set them apart. In Rebuild, Asuka's surname has changed from Sohryu to Shikinami. While there's no official word on why her surname was changed, some theorize that it's to match Rei and Mari's surnames (Ayanami and Makihami respectively). Regardless, I think having a different surname for the Rebuild Asuka is a good idea since her character changes dramatically in Evangelion 3.0.

Shikinami is far more anti-social in the film series. She's often seen with a Wonderswan handheld console, frequently playing it to distract herself from having to deal with the people around her. Sohryu did binge on video games after becoming depressed but was generally more outgoing, revelling in the attention given to her. Both versions of Asuka do not like the attention they receive from admirers at school, though Shikinami reacts more violently towards the boys who approach her (we never see anyone actually approach Sohryu, though we see her crushing love letters under her foot).

Hikari attempts to befriend Shikinami and we see that she's taken aback at someone actually wanting to eat lunch with her. Shikinami agrees but awkwardly says she won't be sharing the lunch Shinji cooked for her -- that is, until she sees Shinji giving Rei a homemade lunch; she offers it all to Hikari instead! XD

As for Shinji and Rei, Shikinami has the same general attitude as Sohryu: she grows to have feelings for Shinji and outright dislikes Rei. With Shinji, Shikinami is just as unimpressed as Sohryu when they first meet. They begin to bond when Shikinami feels lonely during the night; she lays down next to him and they talk for a bit about their reasons for piloting the Evangelion. The following day they argue about lunch and Toji makes the same "married couple" remark he makes for Sohryu and Shinji, both of them blushing and awkwardly trying to deny it.

Shikinami sees Rei as a rival for Shinji's affections, just as Sohryu does. She has the same argument in the elevator with her and feels jealous over her attempts to learn how to cook in order to bring Shinji closer to his father. Shikinami then starts to try to learn how to cook herself so that she can be the one to give Shinji homemade meals, cutting herself often in the process. When they argue in the elevator, Shikinami notices that Rei has many cuts on her fingers also and maybe they aren't so different after all.

At the end of Evangelion 2.0, Shikinami confesses to Misato that she grew up as a complete loner, having no interest in being around other people at all. However, after being forced to deal with the other pilots and students, she's started to learn that being with other people isn't such a bad idea. She seems to reach a good place mentally. As Shikinami expresses no desire for Kaji at all (a major change in itself!), her bond with Misato is never destroyed and the two grow close.

Shikinami also replaces Touji as the test pilot for EVA-03. She meets the same fate as Touji, nearly dying when the dummy plug inside EVA-01 commands the Evangelion to bite down on EVA-03's entry plug. Shikinami is recovered and put into quarantine. Ritsuko tells Misato that Shikinami doesn't have any traces of the Angel inside her but they want to keep her in quarantine just in case.

The following film, Evangelion 3.0, takes place 14 years after the events of Evangelion 2.0. Shikinami is now 28 years old but still looks exactly the same as she did when she was 14. When Shinji, who has been in suspended animation for the 14 year gap, comments on Shikinami's appearance, she blames the "curse of the Eva" for her inability to physically age beyond 14. She's also wearing an eyepatch over her left eye. We don't know how she came to wear an eyepatch (she refuses to tell Shinji when he asks), but it's likely not a coincidence that it's the same eye that was injured in Sohryu's battle against the mass production Evangelions.