To name a few: generics, pattern matching, memory safety by default (though not concurrent yet), no untyped-nil, the capability for C-like performance, deterministic memory management. Protocols are pretty similar to traits as well.
Though the way all of those work is substantially different. Swift doesn't monomorphize generics like Rust does[0], for example.
Swift doesn't have a concurrency story yet, so it is possible to have data races if you share data between threads. An ownership system[1] is in the works, but it is not complete yet.
[0]https://www.reddit.com/r/rust/comments/7gkiie/implementing_s...
[1]https://github.com/apple/swift/blob/master/docs/OwnershipMan...