Rust was inspired heavily by DayZ and originally included zombies. However, the developer decided that the player base in Rust was dangerous enough and removed them.
DayZs development has been slow, I should imagine that this is primarily due to Dean Hall using an engine that was not fit for purpose, and it had to be heavily modified. Unlike Rust.
So basically, they are very similar games, but Rust seems to be slightly ahead of Dayz in their roadmap. If you look at the DayZ Epoch mod, that is closer to how DayZ will end up.
They both have the potential to be great games, but I decided to bet my money on the DayZ pony.