One thing that initially confused me was why both homomorphic encryption (HE) and differential privacy (DP) were mentioned, since I would have thought that the point of using HE would be to get complete privacy (but then I only have a very superficial understanding of either concept). Reading the Apple blog, I see that seems to be because they've sharded the database they're comparing to, so some information is leaked by the selection of which shard to run the computation on, hence the use of DP.
Overall, the blog seems like it may be too vague on the details for anyone to really assess the soundness of the approach, though i don't personally know enough for any more details to do *me* any good.