I haven’t sorted out the extent of it yet, but it seems that most of the reason their personalities are different is that Nao isn’t a perfect mental copy of Sunao - only a perfect physical copy. Her thoughts and memories are her own. IIRC there is some transference - when Nao is summoned, she doesn’t have to catch up with Sunao - she knows basically what’s going on currently. But there are missing details - for instance, it appeared that Nao wasn’t even sure who Hayase-senpai was at first, while Sunao knew exactly who he was and had apparently had some sort of confrontation with him already.

And it appears that Sunao doesn’t know anything about Nao - that all she gets is whatever she’s told.

Which as I think about it, oddly puts Nao in a sort of stronger position - Sunao shares at least some details with Nao essentially automatically, but Nao doesn’t seem to share with Sunao at all - the only way Sunao knows what’s going on in Nao’s life is if she tells her.

I assume that sort of thing is a lot of what it’s going to get into over time. Even though they’re nominally the same, there are actually a lot of differences between the two.