Rust's core object semantics are very nearly that of C. Really, the only major difference between Rust and C is that you can't violate mutable aliasing rules in Rust, even in unsafe, and C has a strict aliasing mode that Rust can't opt into.

The main practical difference is that Rust pushes you away from UB whereas C tends to push you into it; signed integer overflow is default-UB in C, while Rust makes you go out of your way to get UB integer overflow. Furthermore, the general design philosophy of Rust is that you build "safe abstractions" which might require unsafe to implement, but the interface should be impossible to use in a way which doesn't cause any UB. It's definitely questionable how many people actually adhere to those rules--some people are just going to slap the unsafe keyword on things to make the code compile--but it's still a pretty far distance from C, where the language tends to make building abstractions of any kind, let alone safe ones, difficult.