C++, despite RAII, is still manual memory management IMO

How is memory management with Vec and Box in Rust any different from std::vector and std::unique_ptr in C++?

They aren't :)

The main thing I guess is dealing with C APIs in C++. You do have to use C APIs in Rust as well though, but maybe dealing directly with plain C APIs is less common because of how annoying it is. The only way to avoid that pain is to make a Wrapper library that exposes a more idiomatic and possibly "safe" interface. C++ can wrap C APIs just as well though.

I agree with you though, C++ has been doing RAII for a while and I'm not sure why people seem to think Rust invented it. Idiomatic C++ APIs and code are just as "automatic" as Rust for the most part!

C++ doesn't have the borrow checker of Rust. Isn't that what made Rust famous?

(Your literal question is a straw man: no, those aren't different, but Rust offers more.)

I don’t really understand what the borrow checker has to do with the automatic/manual memory management dichotomy.

Even though ownership and borrowing are theoretically orthogonal, they're both needed to make a complete safe system (the context was about managing memory for safety, not merely having a syntax sugar for malloc/free.)

Unique ownership prevents leaks and double-free, and borrow checking prevents use-after-free. Borrowing is necessary to temporarily relax the exclusivity of uniquely-owned objects and access their interior. Access to interior of a unique_ptr (and shared_ptr) decays it to a reference or a bare pointer that is not managed by the smart pointer any more, and can be left dangling. Borrow checker eliminates the manual "be careful not to leave dangling pointers" part.