what are the benefits to using it in Godot?

Primarily performance and type safety. You can get some type safety with typed GDScript, but it doesn't really compare to Rust. You also get all the language constructs for free and access to the entire Rust ecosystem, as you can depend on basically any crate you like.

Does the game run any faster than if it were programmed in CO# or GD script?

Yes, you can get more performance out of your CPU-intensive code compared to GDScript. If you have complex calculations or a large amount of data to process, you will fare better with Rust. The interoperability between Rust and GDScript is also quite good, so you can mix and match as much as you like. I can't really say how it compares to C#, as I have never tested it myself.

Is web more stable with it?

I'm not sure what you mean by more stable, but web exports are possible, which is not currently supported with C#. Rust also does not rely on a runtime like C#, which should give you a smaller export size compared to C#.