Let $X\subset G$ be a subset of a topological group $G$. How are $cl(\langle X\rangle)$ and $\langle cl(X)\rangle$ related in general? Are there results giving conditions on equality of the two?
Here $\langle X\rangle $ is the subgroup generated by $X$ and $cl(X)$ is the (topological) closure of $X$ in $G$.
I don't quite understand what condition you're looking for, but I can give a good example in which these groups are the "least equal" as possible. Note that always $\left<X\right>\subseteq \left<\text{Cl}(X)\right>\subseteq \text{Cl}(\left<X\right>)\subseteq G$. Below is an example in which the first and last inclusions are equalities.
Take $G = S^1 = \{z\in\mathbb{C}:|z|=1\}$ with the multiplication from $\mathbb{C}$ and $X=\{e^{2\pi i \alpha}\}$ a point, for some irrational number $\alpha\in\mathbb{R}$.
In this case $\text{Cl}(X) = X$ and since $\left <X\right>$ is a dense subgroup of $S^1$ the closure is everything. In other words, on the left side you get just the subgroup generated by $X$ and on the right side you get the entire group.