Is there a function which belongs to $H(\mathrm{curl},\Omega)$ but does not belong to $(H^1(\Omega))^3$ ? Similarly is there a function which belongs to $H(\mathrm{div},\Omega)$ but does not belong to $(H^1(\Omega))^3$. If so could you give some concrete example?
Where
$H(\mathrm{curl},\Omega)=\{{\bf u} \in (L^2(\Omega))^3 | \nabla \times {\bf u} \in (L^2(\Omega))^3\}$
$H(\mathrm{div},\Omega)=\{{\bf u} \in (L^2(\Omega))^3 | \nabla \cdot {\bf u} \in L^2(\Omega)\}$
Details about these spaces can be found in chapter 3 of the book "Finite Element Methods for Maxwell's Equations" by Peter Monk.
Let $ u = (u_1,u_2, u_3)$, where $u_i \in L^2(\Omega)$ and $u_i$ is independent of $x_i$. Then $\nabla \cdot u = 0$: I assume that $\nabla \cdot u$ is defined in distributional sense. So for any $\varphi \in C^\infty_c(\Omega)$,
$$(\nabla\cdot u) (\varphi) := -\int_\Omega u\cdot \nabla \varphi = \sum_i\int_\Omega u_i \frac{\partial \varphi}{\partial x^i}.$$
For each fixed $i$, one can apply Fubini to conclude $\int_\Omega u_i \frac{\partial \varphi}{\partial x^i} = 0$. Thus $\nabla \cdot u$ is in $L^2$. However one can choose some $u_i\notin H^1$, so $u\in H(\mathrm{div}, \Omega)$ but is not in $(H^1(\Omega))^3$.