I've only seen up to the second movie. However, if i was to make a guess, it would be what Kaji gave to Gendo.
In the original one, it was a Adam's Embryo, while in Rebuild it's Nebuchadnezzar's Key. I'd say its discovery (before the first movie) is what would be turning point, but a noticeable difference didn't occur until Kaji went to deliver it to Gendo.
In the original series, Gaghiel attacked the UN Pacific Fleet searching for Adam's embryo, but that never happened in Rebuild. Rather, Ramiel became the new 6th Angel, while Gaghiel never appears. Instead, we have the unnamed Third Angel which, if you account for the fact it was subjected to experiments, is reminiscent of Gaghiel.
Also, Kaji released the Third Angel to cover his escape with Nebuchadnezzar's Key, so it's possible that the facility that the Third Angel was in was where Gaghiel attacked before the first movie. Gaghiel may have been captured and experimented on, and became the Third Angel we see in the second movie (thus making Sachiel the Fourth Angel). Kaji knew that by releasing the Third Angel, the facility will do everything they can to prevent its escape from the facility, and in the end, they destroy it.
I'd also note that Sachiel, Shamshel and Ramiel are very much the same since the first movie was more of less the same as the original (very little deviation), except their numbers have been pushed up thanks to the Third Angel appearing in the second movie (and numbering seems to be based off appearance) - from the Seventh Angel on they are different.
While Bardiel is just demoted in number, Asuka is the one who pilots the Eva, not Toji, and the Dummy Plug is over top Shinji and talks backwards which didn't happen in the original.
Jon Lin's Answer talks about Lilith which I totally forgot about. However, both Rebuild's references to Lilith in the first movie and the original series references to her would still occur after the item Kaji gave to Gendo was discovered (Original = "Adam's Embryo", Rebuild = "Nebuchadnezzar's Key"), and as such I stand by my speculations.