Yes, it really needs "are" in both places.
(I guess you could alter the word order a little bit: "These are apples, but not those" would be OK, but it doesn't feel like it matches the Hungarian sentence as well as the solution they suggest here, and it just doesn't sound as good to me anyway. I'm not really sure why.)
I think the reason is the same as in Hungarian. It would mess up the logic of the comparison. You can see it especially if you remove the "but". "These are apples, not those" - is kind of saying the same thing but from a different perspective. The sentence "These are apples, but not those" would be an unfortunate mutation of the two correct ways:
Q: Are all of these fruits apples?
A: No, these are apples but those are not. - Nem, ezek almák, de azok nem.
Q: Which ones are apples? Those ones?
A: No, these are (the) apples, not those. - Nem, ezek (az) almák, nem (pedig) azok.
Note, there is no "but" in this version.