Given $\Omega$ a bounded regular open set in $\Bbb R^d$ we consider
$C_c^\infty(\Omega)$ the space of smooth functions compactly supported in $\Omega$. For $1<p<\infty $ Let's denote by $W^{1,p}(\Omega)$ be the standard Sobolev space. It is common to denote the closure of $C_c^\infty(\Omega)$ in $W^{1,p}(\Omega)$ by $$W_0^{1,p}(\Omega):=\overline{ C_c^{\infty}(\Omega)}^{~~~~W^{1,p}(\Omega)}$$
Note that, by the trivial zero extension the space $C_c^\infty(\Omega)$ can be seen as the space of smooth functions in $\Bbb R^d$ which are compactly supported in $\Omega$ and hence it can be seen as subspaces of $W^{1,p}(\Bbb R^d)$.
I would like to compare the spaces $$W_0^{1,p}(\Omega):=\overline{ C_c^{\infty}(\Omega)}^{~~~~W^{1,p}(\Omega)}$$ and $$\overline{ C_c^{\infty}(\Omega)}^{~~~~W^{1,p}(\Bbb R^d)}$$
More generally if $0<s<1$ is the fractional oder how can we compare the following fractional Sobolev spaces?
$$W_0^{s,p}(\Omega):=\overline{ C_c^{\infty}(\Omega)}^{~~~~W^{s,p}(\Omega)}~~~ and ~~~~\overline{ C_c^{\infty}(\Omega)}^{~~~~W^{s,p}(\Bbb R^d)}$$
Definition: $W^{s,p}(\Omega)$ is the space of class of functions $u$ in $L^p(\Omega)$ such that
$$[u]^p_{{W^{s,p}(\Omega)}}:=\iint\limits_{\Omega\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}} dxdy<\infty. $$ Which turns out to be a Banach space endowed with the natural norm $$\|u\|^p_{W^{s,p}(\Omega)}=\|u\|^p_{L^{p}(\Omega)}+[u]^p_{{W^{s,p}(\Omega)}}$$
NB As the second question might less obvious good answer to this may deserve some bounty accordingly. But I will be okay if one just answers the first question.
These two spaces are the same
This is Corollary 1.4.4.5 in the book 'Elliptic problems in nonsmooth domains' by Grisvard. The proof uses a density result that is claimed but not proven in the book. The density result (in more general form) is Theorem 6 in
A. Fiscella, R. Servadei, and E. Valdinoci. Density properties for fractional Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 40.1 (2015), pp. 235–253. doi: 10.5186/aasfm.2015.4009.