Both are not separable: any countable subset of $S_\Omega$ has an upper bound. Therefore, both are not second-countable. $S_\Omega$ is first-countable: for any $x$ such that $\{x\}$ is not open ($x$ has no predecessor) the countable collection of intervals $(a,x+1)$ for all $a<x$ is a basis at $x$ ; but not Lindelöf: $(a_0,x)$ has no countable subcover. $\overline{S}_\Omega$ is not first-countable: the added point has no countable local basis (for the same reason: the supremum of countable numbers of infima of open sets will be less than $\omega$ ); but is Lindelöf as it is even compact. This illustrates that a subspace of a Lindelöf space may be not Lindelöf.