OK, this is a long one...
I'm not a Japanese speaker, so I'm probably not going to be able to catch all the dialogue changes, unfortunately.
It sounds like all the dialogue has been revoiced, so I'm not going to bother listing that. I'm also not going to bother with the sound levels, as they're almost all different.
This is from Mangle's version of EoE, not the remastered.
First scene: cicadas and waves. In EoE, the waves are much more pronounced.
When Misato is in her car, contemplating the instrumentality project and stuff, in EoE there is a helicopter in the background which is missing in Rebirth.
Also, Misato's line about the Committee and Adam sounds different. It may just be the way the VA says it in EoE, but sounds like different dialogue.
Seele 12, during the Seele/Gendo/Fuyutsuki meeting, sounds like he's saying different dialogue.
The line "Through the sacrament of death, God, and humanity, all living things will be united and reborn as one", sounds like its delivered by a different VA in EoE.
There is a different sound effect for when the Seele monoliths fade out.
When Rei wakes up in her bed, she looks toward the moon. While the camera is panned on the moon, Rebirth has a SE of bedsheets rumpling that EoE does not.
Also, the door opening SE is different when Rei exits the room while the camera is still on the moon. But when the camera is on the door, SE is the same.
Next scene, Shinji is on his bed listening to his SDAT. EoE has a siren and a car SE in the background, Rebirth has nothing in the background.
The scene with the panning up of where Misato is hacking away, the background "machinery" sound is different, plus in EoE you can hear keys clicking.
Also, even after the next few cuts, the clicking of the keys are not the same.
When Misato's screen gets filled with "DELETED" (she thinks she got caught), the SE is different, Rebirth has like a "processing" sound while EoE sounds like klaxons
When the red lines in the room Misato is hacking in fades, after she says "It's beginning", in EoE there's an extra sound effect similar to a large door closing.
During this bit:
the flashing of the communication lines has a different sound effect. Rebirth has a high pitched sound dropping in pitch, while EoE had a constant pitched beeping.
During the whole Magi being invaded thing on the command bridge, EoE has background music playing (I dunno the name), Rebirth does not.
When the display is showing the 5 other Magi's hacking (lots of flashing blue lines)
Rebirth had a TNG-ish sound effect, EoE had no SE here.
At this point, Rebirth starts the same BGM that EoE had going earlier and EoE's BGM stops.
Scene: Ritsuko's cell. In Rebirth, there's some woman blabbering on the PA about "Level 1 Alert". In EoE, the dialog is barely audible (had to turn sound way up to hear).
The SE for the door opening into Ritsuko's cell is different.
When Misato takes the elevator up to the command bridge, there is a short klaxon sound in Rebirth when the elevator arrives at the top.
When Ritsuko is inside the Magi, she says "Am I doing something stupid?". This line is delivered earlier in EoE, which produces a short pause before she lifts her glasses up.
Misato is on the bridge drinking a cup of coffee, right as she asks "How much longer", in EoE you can hear her clothes shift.
After Misato says "They're not going to let us off that easily", in Rebirth, the background chatter is different. EoE has a womans voice mixed in with the background.
Next cut, when Fuyutsuki is talking into Gendo's ear, "and take the 2 Evas by force", Rebirth has a telephone ringing in the background.
But when the shot changes so both Fuyutsuki and Gendo are facing the camera, you can hear the phone in EoE, it's the same SE.
When we see this:
Rebirth has a dinging sound everytime one of the numbers change as well as the previous SE.
EoE has a "warp engine but higher pitched sound" constantly going throughout this cut.
Next cut, Ritsuko is setting her laptop inside of the Magi, in the background there is a computer repeating the same word. EoE and Rebirth has a different word repeated.
Next scene, Seele Monoliths, the line "We'll have to cease contact with the Magi" is delivered by a different VA.
When the JSSDF soldiers are standing up from the grass,
then there is a fast pan to the left, EoE has VToL engine in the background near the beginning of the scene.
Rebirth doesn't have this sound until the scene changes. Also, the engine sounds are slightly different.
Tanks moving along the road. Different engine sounds.
Trucks are driving by, there is a red light somewhat stationary. EoE has a repeated whistle sound after each truck passes. Rebirth has a whistle sound unrelated to the trucks.
The next scene is completely different between EoE and Rebirth. 2 tanks firing vs. 3 tanks firing.
Rocket Launcher SE is different
When, in the command center, the observation stations are getting replaced with "SIGNAL LOST", in EoE there are at least 2 voices in the background, Rebirth only has 1.
Immediately afterwards, there is a pan right of the command center. In Rebirth, there is an extra beeping sound that kinda sounds like a fart.
This lasts throughout the scene in the command center.
Also after the pan, cut to Fuyutsuki saying something with Gendo in the chair, in Rebirth there is a phone ringing in the background. This lasts throughout the scene as well.
Rocket launcher SE is different.
Nerv guy getting stabbed in the back, in Rebirth, as soon as the JSSDF soldier put his hand over the guy's mouth, he starts mumbling/struggling sounds.
The bullets hitting the truck (before it explodes) is different. Rebirth sounds like pings (weaksauce).
The cut with the "KEEP OUT"
written on the walls with a red flashing light to the right, EoE has klaxons going off, Rebirth is this light beeping.
At this point, EoE starts some BGM (dunno the name, low keys on a piano it sounds like).
The cut to the command bridge, panning upwards. Rebirth starts the same BGM from EoE just seconds before.
During this whole bridge scene, EoE has a lot of frantic voices in the background. Rebirth doesn't but it has a phone ringing sound every once in a while.
Misato is over Hyuga's shoulder, says to hide Eva-02 in the lake, you can see reflected flashing in Hyuga's glasses. EoE has some computer noises.
Rei soaking in LCL, Rebirth has a "refrigerator-ish" hum in the background, EoE doesn't have any hum.
Scene switches to bridge, Misato wants to deploy Eva-01 to the geofront. Rebirth starts some BGM, none in EoE yet.
Aoba says that Shinji isn't there, turning around in his chair. EoE starts some BGM, different from Rebirth, Angel Attack I think.
When all the doors are getting sealed, the door closing sound is different.
Back to the command bridge, EoE has frantic voices in the background, Rebirth doesn't.
The sound that these things make is completely different. In EoE, it sounds like they just took off, in Rebirth it sounds like a passenger jet at cruising altitude.
When one of these things turns on its light pointing inside one of those tunnels, the light coming on sounds different. Note, BGM is still different.
Misato is over a display of all the sections of Nerv as they're being replaced with "CAPTURED", Rebirth has a constant warning sound while EoE has a beep for each section getting captured.
During the first bakelite scene, Rebirth just has the sound of bakelite, EoE includes frantic talking in the background and the ongoing siren.
The cut where the bakelite is rising above a dead Nerv guy on the steps, Rebirth starts to have the frantic talking in the background.
Cut to Shinji under the steps, BGM stops on both Rebirth and EoE. But in Rebirth, you can still hear gunshots in the background.
In EoE, gunshots in the background isn't heard until "All non-combat personnel should avoid engaging the enemy", but only briefly.
Back to the bridge, Misato cocks her gun and tells Hyuga to take care of things. Rebirth doesn't have frantic shouting in background, but does have it in next bridge scene.
Hyuga and Aoba on the bridge, "They've probably been planning this from the start", Rebirth has no shouting in the background but has telephone ringing.
After the JSSDF blows up the entrance to command center, Rebirth has shouting in background again and still that phone ringing. EoE has no phone.
Shinji under the stair case, over the radio you can hear "Yanagihara and Shinjo units, proceed to lower level" in EoE. In Rebirth, this line is completely different.
(The fansub translates it as: Special Team charge into the mechanics room)
Misato running down the hall shooting, her gun sounds very different between EoE and Rebirth
When Misato is tuning a JSSDF radio, the dialogue is slightly different. EoE: Roger that. We have the purple unit surrounded. Rebirth: What is the situation of team 7?
In Rebirth, they also mention Eva-02 being at the bottom of the lake, in EoE this bit is inaudible, you can hear someone talking but can't make anything out.
N2 mine explosion is slightly different at first.
Rebirth doesn't have that static sound when the shockwave hits the Nerv pyramid (Rebirth doesn't have sparks coming off the pyramid). As well as that extra explosion.
During the scene in the Eva graveyard, with the occasional flash of 2nd impact, Adam, angel DNA, etc... Rebirth has this almost pipe organ-ish BGM (Infantile Dependence, Adult Dependency), EoE has no BGM here. This BGM is still on when the scene cuts to inside Misato's car.
Scene in office in Tokyo-2 with that swinging pendulum. EoE has a lot more bangs and creaks in the background than in Rebirth.
Scene of the big hole in the geofront, someone says "Surface heat has dissipated". In EoE there's this loud rushing water like sound. Rebirth doesn't have.
Back to the JSSDF commander. In Rebirth there is a helicopter like sound in the background, EoE has people talking on the radio in the background.
Asuka's "I don't want to die". In Rebirth, she starts off saying it at an audible level, whereas in EoE she's almost inaudible (whispering) at first.
Also, in Rebirth, about 30 seconds into Asuka's chanting, there are sounds in the background like a baby crying and other voices (lots of them near the end).
In EoE, you hear Kyoko talking to her, and maybe an extra voice right before Asuka shouts.
In Rebirth, near the end of Asuka's chanting, BGM starts (the one that plays during instrumentality, choral stuff).
After Asuka screams and after the flashing images, cut to the waves on the shore, now in EoE the BGM starts (same as Rebirth earlier).
Eva-02 lifting the ship out of the water, Eva-02 growls in EoE.
The sound of the ship breaking apart (in front and back) is different.
Eva-02 throwing the ship, Eva-02 is making some noise in EoE.
After the 2 huge missiles explode on Eva-02, the scene is filled with black smoke then Eva-02's head bursts through the smoke. EoE has a ripping sound effect here.
The cut to the VToL (or whatever it is) right after Shinji sulking in Misato's car. The VToL is firing missles but Rebirth's SE sounds like a machine gun, EoE sounds right.
The 5 minutes of power left warning sounds different (Rebirth is higher pitch).
At this point, the BGM isn't synced up with the action between EoE and Rebirth, until the VToLs come out.
When the last VToL comes out behind Eva-02 (she's about to kick it), its engines are silent in Rebirth (they must have forgot to add the sound?), sounds fine in EoE.
The sound being made as the MPEs are about to separate from the planes is different.
As the Kaworu plug is inserted, in EoE you can hear a growling sound (MPE?)
When the MPEs spread their wings, in Rebirth it sounds mechanical, in EoE it sounds organic (like a bird).
As the flock of MPEs fly down into the hole on top of the Geofront, in Rebirth they sound like a bunch of airplanes, in EoE this sound is similar to a pod racer from ST-ep1.
As the MPEs circle Eva-02, in Rebirth, the ending music starts. In EoE, they continue to make that pod racer sound.
I'm sure I missed a lot of other small things. I listened to these with headphones so other noises wouldn't bleed in, but sometimes there's too much going on. These descriptions probably aren't good enough unless you've got a really good memory or if you're watching EoE/Rebirth. I should have had the foresight to mark the timestamps of when each of these things happened in EoE and in Rebirth. Maybe I'll do that later.
Is that true? (If so, then one difference I spotted was the 25 "your decision alone/the world you wished for" empty audtiorum scene BGM playing when Misato told Shinji what the Angels are. As well as the scene with Asuka saying "I don't wanna die", there seemed to be different music playing there, and I think the voices were either different or one was missing. 
Rebirth
EoE
Rebirth
EoE
Rebirth
EoE
Rebirth
EoE
Rebirth
EoE
