The Accidental Tech Podcast had an interview with Chris Lattner where he discussed the future of Swift as a systems language and compared it to Rust. Rust has a very upfront memory ownership model that requires programmers to be explicit about memory management. This allows Rust to have great performance and allows the compiler to ensure memory is used safely that is not an option with C.
With Swift, either you pretty much don't think about memory (it uses Automatic Reference Counting so you only need to care about cycles), or you need to go down to C-style memory semantics with the various Unsafe constructions. There are cases where you could get much better performance because the programmer knows the lifecycle of the objects being used, but that can't currently be expressed in Swift. It can be expressed in Rust.
To be a good systems programming language, Chris said that Swift will need to create a memory ownership model (and mentioned Rust as having ideas that might apply). He would like that ownership model to be opt-in for specific pieces of your code that require it: most people could use ARC, while people that need performance in a specific piece could be more detailed about the memory management. It's on his list of things that Swift will acquire over the years so it can achieve world domination.
So there are really pretty good reasons that Mozilla put together Rust. The browser is probably the most widely exposed attack surface right now, and the history of buffer overloads means there needed to be a safer way to code.