The two languages are quite different. Go is billed as a "systems language", for whatever that means, but in practice occupies a space similar to Java. It's quite well-suited for server and applications programming, but various design choices mean that it's less well-suited for lower-level programming domains like operating system components or device drivers, performance-intensive video games, or hard-real-time embedded systems.
On the other hand, Rust is aimed at being a replacement for C++ or even C: it allows for precise control over lower-level aspects of the machine while providing more static guarantees of correctness by means of a powerful type system. For example, Rust prevents dangling and null pointers and disallows access to uninitialized memory while retaining manual memory management, as well as allowing garbage collection as a library-provided feature and not a core language feature. In that sense, Rust is suitable for problem domains that Go is not.
On the other hand, Rust is aimed at being a replacement for C++ or even C: it allows for precise control over lower-level aspects of the machine while providing more static guarantees of correctness by means of a powerful type system. For example, Rust prevents dangling and null pointers and disallows access to uninitialized memory while retaining manual memory management, as well as allowing garbage collection as a library-provided feature and not a core language feature. In that sense, Rust is suitable for problem domains that Go is not.