There is an important difference: in Rust you can write a new trait, with new methods, and impl that trait for a type without having to change the existing structs/enums which comprise that type at all. You can also do this (with some restrictions, "the orphan rule") for types defined in another library.

In C++ classes, you have to be able to modify a class definition to extend it with new methods (or a new ancestor to multiply inherit from).