I hear this a lot, but there are many things for which you can use both, so they're competitors. Also most Rust developers want to use it everywhere, including areas where you normally reach out to C, so if people that like Rust want to use it for things that are usually made with C, and Zig is a replacement for C, it seems to me they're competitors, no?
Zig is a better C. No abstractions. Close to bare metal.
Rust is a better C++. Abstractions to simplify application level programming.