We can make a metric space $X$ which is a subspace of $\mathbb R^2$ where $\overline{B_r(x)}\neq B_r[x]$. For example we can take $X=\mathbb R^2-\{(x,y):0<x<1\}$. Consider $(0,0)$ and $(1,0)$ in $X$. Take $\overline{B_1(0)}$ and $B_1[0]$.See that $(1,0)\in B_1[0]$ and $(1,0)\notin \overline {B_1(0)}$.
See that $(1,0)$ is not a limit point of $B_1(0)$. I think this is a more lively examples than unit balls of discrete metric which seem quite artificial. This situation is easy to visualize why the inclusion may be proper. Can someone provide me with some information as what further things related to these I can look into. For example I have seen that if for every ball, closure of open ball is corresponding closed ball in a metric space $X$, then open balls are connected.
In $\Bbb R$ there are also easy examples, like $X=[0,1]\cup \{2\}$, where $2 \in B_1[1]$ but $2 \notin \overline{B_1(1)}$. Of course always $\overline{B_r(x)} \subseteq B_r[x]$ for any $x \in X, r>0$.
It's fun to point out that it can happen, as many intuitions are built on the Euclidean space, where we have convex balls, where this phenomenon doesn't occur. But it's not an important thing IMHO.