The fight between Aziraphale and Crowley about Gabriel goes like this:
Aziraphale believes that it's Safest for them to deal with the gabriel situation by being involved and solving it together.
Crowley believes that it's Safest for them to get as far away from the Gabriel situation as possible, either by removing gabriel from them or themselves from him.
Aziraphale also believes that it's Right to help Gabriel, because Aziraphale believes in the innate goodness of everyone and everything.
Crowley believes that it's Right to not help Gabriel or frankly to care about him at all, because Gabriel hurt Aziraphale repeatedly over millennia and then tried to destroy him.
They approach this issue with such a difference in their core beliefs and values that they have to disagree, it's inevitable. And that would be fine. Except that the way they resolve this argument is utter bullshit.
First off, Aziraphale responds to Crowley with an ultimatum: do it my way or leave. (Which, hello, trauma from being cast out of heaven? Wtf Aziraphale. )
Crowley responds by telling Aziraphale he'll have to do this alone, and then predictably leaving.
Crowley only changes his mind and returns because he learns that Aziraphale is in real existential peril. Then they do a series of Things, none of which actually revolve their big disagreement.
Crowley offers a sort-of apology ("can we take it as stated?") but doesn't say what he's apologizing for. He doesn't tell Aziraphale what he thinks he got wrong (spoiler: because he absolutely doesn't think he Was wrong, and his belief is reinforced by the threat of destruction hanging over Aziraphale's head). It's "I'm sorry for whatever I said, get in the car" all over again.
Aziraphale doesn't even apologize lmao, and why should he? He must be right, because Crowley's back! And Crowley's doing their ritualistic "apology" dance that they use in place of actual fucking conversation. So Aziraphale doesnt need to acknowledge anything about his line of reasoning or his belief system, they can just carry on.
Especially because, crucially, Aziraphale doesn't know that Crowley was actually right! And helping Gabriel will put them both in extreme danger! And he doesn't know this, because Crowley didn't tell him!
Crowley didnt tell Aziraphale about the Book of Life punishment because he believes Aziraphale is Safest not worrying about that, and that it's Right for him to protect Aziraphale from that hurtful, harmful knowledge.
They literally do. not. discuss this issue ever again. They have no idea, or none that we're textually shown anyway, why the other reacted the way they did to Gabriel. They simply move forward with the problem, without even an iota of enlightenment about each other's points of view. They think they already know each other perfectly, right? They've been talking for millions of years, Crowley loves rescuing Aziraphale, etc.
So then we get to ep6. And it's really the same fundamental disagreement:
Aziraphale believes they will be Safest with their hands in the game.
Aziraphale believes it's Right to force Heaven, through what he has been manipulated into thinking is his new authority, to do Good (actual moral good, like he and Crowley do).
Crowley believes it's Safest for them both to stay as far away from the machinations of heaven, and by extension hell, as possible.
Crowley believes it's Right for him to reject Heaven as they once rejected him, to reject their whole dichotomous system in fact. He believes it's Right for Aziraphale to reject Heaven too, because Heaven tried to destroy Aziraphale.
And once again, Crowley doesn't tell Aziraphale of the danger they're in - Armageddon the Sequel, plus Archangels being demoted and having their memory wiped.
They still do not and cannot understand one another's motives and beliefs here. Because they have never had this freedom of choice before these last few years, and in that time they have never talked about this openly. Weirdly, Gabriel's arrival gave them a chance to hash all this out. In another world, maybe they knew exactly what the other wanted, and could therefore choose each other at this critical juncture.
But that's not what happened.
(cue my "the irony of the serpent of Eden protecting someone he loves from knowledge" tag).
Edit to say, I have Thoughts about Why they communicate like this, which I'm trying to gather for another post.