Consider the shift space $X \subset \{0,1\}^{\mathbb{Z}}$ obtained by forbidding the words $1 0^m 1^n 0$ for all $m, n \geq 1$, and denote the shift map on $X$ by $\sigma$. Since a point of $X$ can contain at most three transitions from $0$ to $1$ or back, $\omega(\sigma)$ consists of the two uniform points (all-$0$ and all-$1$), and in particular it is closed. However, $x = \ldots 0 0 0 1 1 1 \ldots$ is nonwandering because arbitrarily close to it we find the point $\ldots 0 0 0 1^n 0^n 1 1 1 \ldots \in X$ for some large $n \geq 0$ that returns close to $x$ after $2 n$ shifts. Hence the closure of $\omega(\sigma)$ is properly contained in $NW(\sigma)$.

Consider the shift space $X \subset \{0,1\}^{\mathbb{Z}}$ obtained by forbidding the words $1 0^m 1^n 0$ for all $m, n \geq 1$, and denote the shift map on $X$ by $\sigma$. Since a point of $X$ can contain at most three transitions from $0$ to $1$ or back, the only $\omega$-limit points of $X$ are the two uniform points (all-$0$ and all-$1$). However, $x = \ldots 0 0 0 1 1 1 \ldots$ is nonwandering because arbitrarily close to it we find the point $\ldots 0 0 0 1^n 0^n 1 1 1 \ldots \in X$ for some large $n \geq 0$ that returns close to $x$ after $2 n$ shifts. Hence $\omega(\sigma)$ is properly contained in $NW(\sigma)$.