First, to establish what we're not losing by using Rust:
* It can be as fast as C++ (in essentially all cases--all but compile-time metaprogramming related situations, after some in-the-works reforms are implemented).
* It can be as deterministic as C++ (usable in hard realtime systems).
* It has great FFI-level C compatibility.
Advantages over C++:
* It is substantially safer than C++. Outside of unsafe blocks, it is memory-safe and many other forms of undefined behavior are absent, including data races (unlike not only C++, but Java, Go, and most other mainstream languages that support multithreading).
* In a number of cases, it could potentially be faster than C++ by utilizing Rust's much stricter aliasing semantics.
* It should be able to compile much faster than C++, and it can infer much more thanks to its type system.
The above are things that I don't think can be shoehorned into C++ backwards-compatibly. I'm not including features like ADTs because I see no reason C++ couldn't have them.
There are plenty of reasons not to choose Rust over C++, including language immaturity, implementation immaturity, library immaturity, ecosystem immaturity, and lack of familiarity. But language feature for language feature, it offers substantial benefits.
* Pending some in-the-works reforms, and not including implementation-specific features like labels as values (https://gcc.gnu.org/onlinedocs/gcc-2.95.3/gcc_4.html#SEC64).
Incidentally, some of those Rust programs are written for multicore:
binary-trees Rust ≈ CPU Load 96% 86% 92% 92%
fannkuch-redux Rust ≈ CPU Load 100% 100% 100% 100%
regex-dna Rust ≈ CPU Load 59% 79% 80% 59%
Rust was designed to be memory-safe. That means, unless you explicitly mark a section of code as "unsafe", it isn't possible to have segfaults, have buffer overflows, use memory after freeing it, etc. A very significant amount of software bugs - famously, Heartbleed, but also a huge proportion of other security bugs in everything from servers to web browsers - can't happen under Rust code. The point is to have a much more secure programming language, as in resistant to attacks, while still being as fast and as usable as other C-style languages.
The reason Mozilla are developing this is for Servo, a new browser engine which should be far safer than existing ones - although it's also useful for pretty much any other code, as it wipes out entire classes of bugs with not much effort. If you write in a C-like language today, you could pick up most of Rust (lifetimes included) in a couple of days of playing with it.