Hardness of CDH in different groups What is the difference of the CDH problem in different groups? In particular, given a group $\mathbb{G}_1$ of order $q$ that is a subgroup of $\mathbb{Z}_q^*$, $q$ prime, and another group $\mathbb{G}_2$ of order $p$ that is a subgroup of $\mathbb{Z}_p^*$, $p = 2^n$, is solving the discrete log problem easier or harder in one or the other?
If a prime $q$ is large enough discrete logs and CDH in $\mathbb{Z}_q$ are traditional hard problems in cryptography. 

Your other example $\mathbb{Z}_{2^n}$ is typically easy to solve because it can be solved progressively modulo increasing powers of $2$.

May be you intended to consider the finite field with $2^n$ elements $\mathbb{F}_{2^n}$ and not numbers modulo $2^n$.
If so, you might be interested be this related question: How robust is discrete logarithm in $GF(2^n)$?