As was pointed out in that thread, Anima and the Rebuild/NTE films share a number of ideas, with concepts introduced in one later popping up in the other. I thought it might be useful to catalog some of these elements in their own thread so they're much easier to find. I've put together a bunch of images for an easier direct comparison and included the dates for the Anima chapters so you can see when they were published relative to the movies.
Here's an overview of the release dates for each:
1.11: September 01, 2007
2.22: June 27, 2009
3.33: November 17, 2012
Section 1: January 2008-August 2009
Section 2: September 2009-May 2012
Section 3: June 2012-April 2013
Here we go:
In Anima, NERV is taken over by the UN after Third Impact and Instrumentality fail, and it's restructured into NERV JPN. Misato is appointed the new commander. Gendo and Ritsuko are missing, and I don't know Fuyutsuki's whereabouts.
In 3.0, it's revealed that NERV was again significantly changed after Third Impact, splitting off into two groups: Neo-NERV and WILLE. WILLE is made up of most of NERV's old staff with Misato again taking the helm. Neo-NERV consists of Seele, Gendo, Fuyutsuki, Kaworu and the Ayanami Series.
In both continuities, Misato has stepped up into the command role and taken charge of the fight for humanity against Seele, Armaros, and Instrumentality. It's not really clear whether Misato is the supreme commander of WILLE in the same way Gendo was for NERV, but she serves the same function as far as the story is concerned.
The Ayanami Series:
The Next Time preview for Q showed us that infamous image of four Reis all active at once: one teenage Rei many assumed to be Rei III and three loli Reis. None of that bore fruit in Q where we had just one Rei active and 60+ who had since passed on. There probably were multiple Reis at one point, but we don't know for sure.
Anima presents us with four active Rei clones: Rei III (called Trois), Quatre, Cinq, and Six. Quatre, Cinq and Six weren't supposed to have independent personalities, but were all linked to and remotely controlled by Rei III. They started to develop their own identities after Quatre experienced some type of mental contamination. This group is the opposite of the Q group, consisting of three teenagers and one child. They were introduced in February 2008, over a year before 2.0 came out!
In both continuities, Maya steps into Ritsuko's shoes as chief of Eva tech and maintenance. She quite literally steps into Ritsuko's shoes (and clothes) in Anima!
Ritsuko, who is absent for most of Anima, resurfaces in the February 2013 chapter wearing a spiffy green coat similar to the one she wears as Misato's vice captain. Looking at the chapter's date, it's probably a reference to her appearance in Q.
Anima's Mari is a genetically engineered child with cat ears, created by the US to pilot Evangelion WOLFPAC and is close friends with Rei Six. Mari debuts in Section 2 which came out in late 2009, after the release of 2.0. Mari's cat-eared helmet in 3.0 could be a subtle nod to this Mari, or just an amusing coincidence.
Evangelions 0.0 and Mark.09:
After it's destroyed, Unit 00 is rebuilt into four upgraded Evas: Rei III's Unit 00 F-Type, and three nearly identical Evas called Eva 0.0 piloted by her three clones. The three 0.0 units were made from parts of the Mass Production Evas and have backpack units containing the MP Evas' S2 organs.
Two of them are stationed in orbit around the Earth, making them the first Evas to achieve powered space flight:
Mark.09 is also piloted by Rei clones, looks like an upgraded Unit 00, has unlimited power, and comes equipped with a bizarre rocket engine backpack for flight. This is all just a ruse and it's really the Vessel of the Adams, of course, but that's another similarity I'll get to in a minute.
Eva Unit 02:
Unit 02 is upgraded into the Unit 02 Type II following the battle with the Mass Production Evas, but it otherwise bears almost no resemblance to the Unit 02 upgrades seen in 3.0. However, it does take a trip into space in Chapter 12 (Feb. 2009) with the aid of a support unit called the UX-1 Allegorica:
The Allegorica is much more Gundam-esque than the Booster Rocket Unit 02' uses, but it turns the Eva into a quadruped with two fully mechanical prosthetic limbs, similar to Provisional Unit 05.
US Evangelion WOLFPAC and Code 777:
Mari's Eva, WOLFPAC, has a bestial form that's shown off in the February 2013 chapter. Like Unit 02, it grows paws and develops other animal-like traits, and it has a set of restrictors pushing up out of its back. That Eva shadow off to the left is there for scale. WOLFPAC is enormous!
Armaros and Eva Mark.06:
In Chapter 7 (August 2008), Anima introduces its primary antagonist, a jet black Evangelion called Armaros who appears on the moon, much like Kaworu's deep blue Mark.06. Armaros is much bulkier and deviates heavily from the standard Evangelion body design, but when you compare its head to Mark.06's, the resemblance is striking:
I flipped Mark.06's picture in the above image so it's easier to compare their profiles directly. And like the Mark.06, it comes equipped with a Lance (the original LoL), has a new internal power supply ("QR Signum" for Armaros), and it has a halo. The halo is oriented behind the head, however, something not seen again until Eva 13.
From what I understand, Armaros is some kind of "Black Adam" or dark counterpart to the original Adam that rains down untold destruction upon humanity from the moon. While it isn't actually black, Mark.06's true form is a dark-skinned Adam, and it's suspected of bringing ruin upon the Earth with Third Impact.
When we finally catch up with the Mark.06 again in 3.33, it has bloated to an enormous size after fusing with Lilith's headless corpse in its last battle. Eva 13 is easily able to stand on its head and face. In one of the final chapters, Armaros also grows to an enormous size, with each of its arms being at least the size of an Evangelion if not larger. This occurs in one of Section 03's "missing" chapters, but I'm pretty sure it's from the finale in April 2013.
Torwarts and the Vessel of the Adams:
Armaros has two henchman working for it, black Evangelions called "Torwarts" or "Victors." I haven't been able to learn much about them, but the function they seem to serve for Armaros is similar to the one the Vessel plays for Evangelion 13. The horn and crest arrangement on their faces is pretty similar to the one the Vessel has with its new head. It's easier to see the similarity when you look at it from the front:
Update: One of them even fights against Beast WOLFPAC!
It looks like the Torwarts first appeared in the July 2010 chapter.
"Apple's Core" and the Black Moon:
No idea what the object on the left is supposed to be, but it shows up in Section 2 Chapter 14: "The Apple's Core" (Jan. 2011) and looks very much like Lilith's Black Moon from 3.0.
Valley of Headless Bodies and the Failures of Infinity:
Section 02 Chapter 09 (June 2010) features a valley full of giant headless bodies that are petrified. They're said to be from a failed Instrumentality from long, long ago. And of course 3.0 has all of those headless Unit 01 clones laying around, which Kaworu claims are from the aborted Third Impact. Interestingly, the headless corpses in the Valley seem to be very human in shape and proportion, while the FoI are very clearly Evangelions, complete with armor.
New! The Magoroku Exterminate Sword:
Yamashita designed this sword as a special weapon for Unit 01 all the way back in NGE, but it only appeared in toys, video games, and action figures until the trailer for 3.0+1.0 at the end of 3.33, where Unit 08+02 used it to cut through a horde of Mass Production Evas.
As TMBounty_Hunter very helpfully pointed out in the Anima General thread, the sword that appears in that trailer is actually the Magoroku Stage 02, a redesigned version that first appears in Anima Chapter 01 (January 2008). The original model looked like this:
New! Lilith's Barrier:
Anima's Lilith emits a strange black barrier after Shinji stops Instrumentality. It swallows Gendo in the first chapter, but I can't find any mention of it or Lilith after that. 3.0 also has the mysterious Lilin-repelling L-Barrier that surrounds the Tokyo-3 region after 3rd/4th Impacts are averted.
New! The Evolution of Unit 01:
It undergoes a second evolution at the end of Section 2/start of Section 03 (June 2012), transforming into Super Eva Final Stage or Stage: Perfection, seen on the far right.
It looks like a few design elements used in Super Eva and Final Stage may have been reused in Eva 13's design, specifically Final Stage's boxy, squared off thigh armor, the raised flare on the back of the head armor shielding the skull case, and the shape of Super Eva's crest. Final Stage also has a smaller v-shaped abdominal plate right under the chest armor which is present in 13 but not in vanilla Unit 01.
When Shinji triggers Unit 01's awakening in 2.22, it transforms into a Radiant Giant with wings of light. Rebuild's Unit 01 is strongly suspected to be one of the four Adams, so it taking on this form makes sense.
In a really peculiar twist, Super Evangelion also takes on an Adam-like form as a prelude to its evolution in Section 02's Special Comic (March 2012). The Eva starts glowing beneath the armor, grows yellow energy wings, and is encased in some kind of projection that looks exactly like a yellow Adam! It even has light shining from its eyes and mouth like 2.22 Unit 01, though it is yellow instead of reddish pink.
In the January 2013 chapter, Super Eva seems to have an arm and a leg made out of light, very similar to awakened Unit 01's AT Field arm. It regains the Radiant Giant form in the last couple chapters to battle Armaros again, and the Eva's wings appear to lead up into a vortex of light over its head. You can see that form in the Giant Armaros image.
Super Eva is Lilith based, so the Radiant Giant form is very odd. Even at the height of its godhood in End of Eva, Unit 01 only had glowing wings and eyes. The rest of it was dark and meaty, almost the inverse of Adam. So how does that happen?
Kaworu might provide the answer. In the last chapter, he apparently tells Shinji a story about Adam, saying that he removed the flawed, sinful and human part of himself, and implies that Shinji is that missing part. You can read more about that here!
Super Eva and Shinji are still one being in two bodies thanks to Unit 01 having Shinji's heart, so apparently Shinji being a part of Adam means that Super Eva is as well!
And just for fun:
One of the Torwarts steals Shinji's heart from Super Eva midway through Section 2. It's the glowing triangle embedded in the Towart's chest. After that, it becomes Shinji's second body in Super Eva's stead. You could say the Torwart becomes Adam's vessel!
This is everything I've been able to find so far. It's hard to go too far in-depth without knowing what the text actually says! I might need to go back over this later and clean up some of the language. If anyone knows of anything else or spots any mistakes, feel free to chime in!