Please do see: Jon Gjengset's presentation: "Considering Rust" - Considering Rust - YouTube a very reasoned and unbiased description of the pros and cons of Rust and why one might want to consider it.
Personally, as a long time user of C/C++, I'm adopting Rust because:
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Correctness - Rust's type system and memory usage correctness checks prevent be from introducing all kind of bugs that will waste my time after deployment. This also has implications for improving the security of ones code.
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Performance - Rust provides performance that matches C/C++ and sometimes surpasses it.
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Applicability - With no run-time system or garbage collector requirements I can use Rust all the way from embedded micro-controllers to server side to browser (not that I have looked into that last one much yet.
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Ease of use - With Cargo and the Rust crates it is so much easier to make use of all the wonderful libraries people have created. Rust is a much higher level language than C but without all the mind numbing complexity of C++. You don't need to know it all to start getting useful things done. Importantly what you don't know cannot cause you problems as it can in C/C++.
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Community - Rust is a big language. It has features and implements concepts that are likely new and unfamiliar to users of C/C++ (or other languages I guess), the community has proved very supportive and helpful.
If these are things you value then you may well like to consider Rust as well.