Is this problem same as discrete logarithm?
This is exactly the Computational Diffie-Hellman (CDH) problem. Rename $h=g^z$ and $g'=g^y$ in what you wrote, then the problem becomes:


  Given $g$, $g^z$, and $g^y$, compute $g^{yz}$.


See this discussion to address your question about the relationship between CDH and discrete log.

edit: 


  Q1: If we know $(g′)^z$ and $g^z$ then is it easier to find $z$?


No, it is just as hard as discrete log. If you are given just $g$ and $g^z$ then you can generate other pairs with the same discrete log relationship yourself, via $g' = g^r$ and $(g')^z = (g^z)^r$. So having extra such pairs doesn't add any new information.