Rust throws you into a very familiar environment in terms of "EVERYTHING CAN AND WILL KILL YOU, OH MY GOD." For the moment, there are zombies (these are apparently getting pulled out at some point...), animals, and the most terrifying of all: other players.
So what makes Rust stand out from DayZ? Largely the minecraft piece makes it really enjoyable to play. In DayZ, if you died... that was it, you started from scratch all over again and you were lucky if you had some gear stashed somewhere that you could survive long enough to get to. In Rust, you're able to build your own base. Now, keep in mind that bases aren't impervious, and people can generally ruin your day if they're dedicated. You can also craft weapons and weapon mods in what is a pretty simple system (once you get it down...)
If you're into emergent gameplay, PVP, and dealing with a game that is squarely in the alpha phase of it's development, you should give Rust a shot. I find it very enjoyable.