Rust, with the borrow-checker, strong-arms the developer into using memory patterns that the compiler manage handle without the overhead of reference counting or other runtime mechanism.
Those same memory patterns are available in Swift as well, which the compiler can then optimize as well (in theory -- as far as I'm aware there are various unimplemented optimization opportunities in Swift).
So in a sense, in Swift you can opt-in or out of rust-level memory management.
Of course, a big difference is that Rust tells you up-front when you've got it right, but in Swift it's not at all clear. You need to profile. Swift does have a clear delineation between reference types (which use ARC) and values types, so you can get on the right path by choosing value types. Value types in Swift are very rich and powerful so it's not a hard path to follow.
Swift is much higher-level than Rust. It's at best at Go's level. You'd expect Swift's performance to be good, but in low-level tasks Swift is actually closer to JS/TypeScript than anything else: https://github.com/ixy-languages/ixy-languages
It's using some non-optimal patterns that you would not normally use in Swift for code if you wanted it to perform well. (There's a nice performance analysis if you click in, showing that it's using ARC in an inner loop, which is taking 3/4th of the time. That's not something you need to do.)
If your application or backend or whatever can tolerate GC, I would strongly recommend that you use it. While Rust has made manual memory management easier than either C++ or C, it still requires more work than a GC'd language.
Rust has a lot of things going for it, but this obsession with writing everything in Rust that has taken HN by storm is a little confusing.
Swift is unequivocally a higher-level language than Rust and definitely should be used for iOS development unless you have a very good reason to use Rust.