Are the endings shown in the anime series and shown in The End of Evangelion different/alternate or are they same but in different places?
First the history of the endings.
Gainax studio went through a huge financial crisis near the ending of evangelion. They decided to end the series anyway, and went for a low-budget ending (the episodes 24-25).
The television series takes a very avant-garde approach and focuses on a first-person psychological account of the instrumentality process from the perspective of its main characters (primarily Shinji Ikari). ref
The fanbase reaction was brutal. From death threats to graffiti, hatemail and the likes. Anno's take on that:
On Evangelion's last two episodes , which upset many fans... ANNO: I have no problem with them. If there's a problem, it's all with you guys. Too bad.
(emphasis given because of verbal cue given by Anno himself during the interview
Later they decided to do the feature animation "The end of evangelion", retelling the ending.
You have to take in mind that Anno's style of storytelling is very fluid. He would only work on a new episode when the previous was finished:
Toshio Okada: But--the last scenes were never fixed. When I talked to Mr. Anno a month ago, he said he couldn't decide the ending until the time came. ref
Anno comments on the development of the series:
Hideaki Anno: “The development of Evangelion gives me the feeling of a ‘Live’ concert. Whatever the story or the development of the characters, I made them without a plan. During the production, whether listening to various opinions or analysing my own state of mind, I kept questioning myself. I got the concepts from this personal stocktaking [self-assessment]. At first I had intended to make a simple work featuring robots.
But even when the main scene became a high school, it did not differ compared to other productions in the same style. At this point, I did not really think of creating a character with two faces, two identities: one shown at school, and the other inside the organization he belongs to [Nerv]. The impression of ‘Live’ concert that gives me the birth of Eva, was the team joining me in developing it, in the manner of an improvisation: someone plays the guitar and, in response, the drums and bass are added. The performance ended with the TV broadcasting ending. We only started working on the next script once the previous one was done. ref
So they made the film with the new ending, re-telling the story from another perspective. Some classify the first ending as the personal instrumentality inside Shinji's mind, the other as the external instrumentality.
Anno, in the best "death of the author fashion", said (and I am sorry I couldn't find the source) that the "true" ending lies with the viewer. When he started to make the new movies (the rebuild of evangelion), he is simply telling another evangelion story. Like folklore, the true story is nowhere to be found, all of them are true, depending only on the viewer's impression.
Hideaki Anno: ”Eva" is a story that repeats.
It is a story where the main character witnesses many horrors with his own eyes, but still tries to stand up again.
It is a story of will; a story of moving forward, if only just a little.
It is a story of fear, where someone who must face indefinite solitude fears reaching out to others, but still wants to try.
We hope that you look forward to the 4 new retellings of this story.ref
So the truth is that the real ending is all / any of them. The interpretation of what you saw on the screen is intentionally left open.
The endings are actually the same. The series ending shows how humans perceive the human instrumentality in their minds (their loneliness erasing and all their minds merging). The movie ending shows the events that happen in the outside world (the physical events).
Really, the movies were result of pressure from fans. :D
They are different.
In the original series, Shinji ultimately accepts human instrumentality and joins the rest of the human race in the pool of LCL, surrendering his fear, and by doing so also sacrificing his individuality and personhood, to become part of a singular superorganism.
In EoE, he rejects human instrumentality, unable to let go of his fear of becoming one with everyone else. He ends up on a beach, alone except for an unconscious Asuka beside him. Everyone else is gone. Earth is devastated. The chances of rebuilding civilation are slim but not non-existent. At a loss for what to do, Shinji walks over to Asuka and chokes her. She wakes up and expresses her disgust. Credits roll.
I personally like the same ending theory. I feel like while Shinji was in instrumentality he worked through his issue through ep. 26's sequences as instrumentality intends to do, but since Rei is a clone of Shinji's mother and instrumentality happened with him as the center point inside the Eva his mother fused with, he was pulled out of it by Rei/Lillith enough to understand what instrumentality was and rejected it after having experienced it. Choosing to instead endure the pain of attempting to find the connections he's looking for properly, rather than being a single being to have those connections effortlessly.