Everything is in the title, $L=\bigcup_{n\ge 1} \overline{\Bbb{Q}_p}((t^{1/n}))$ is algebraically closed and its elements are relatively easy to understand, whereas $k=\bigcup_{n\ge 1} \overline{\Bbb{F}_p}((t^{1/n}))$ is not algebraically closed (*) so $\overline{\Bbb{F}_p((t))}$ is more mysterious.
Thus, it might be helpful to find a subring $R\subset L$ with a surjective homomorphism $R\to \overline{\Bbb{F}_p((t))}$.
(*) this is because $k$ is purely inseparable over the maximal tamely ramified extension of $\Bbb{F}_p((t))$ whereas the splitting field of $x^p -x-t^{-1}$ is a separable wildly ramified extension
$\color{blue}{\text{Edit}}$ letting $R$ to be the integral closure of $\Bbb{Z}_p[[t]][t^{-1}]$ in $L$ may do the trick. $1/p$ is not integral over $\Bbb{Z}_p[[t]][t^{-1}]$ so $R/(p)$ shall contain $\Bbb{F}_p((t))$. As every monic polynomial $\in \Bbb{F}_p((t))[x]$ lifts to a monic polynomial $\in \Bbb{Z}_p[[t]][t^{-1}][x]$ which splits in $R$ we'd get that the maximal ideals $\mathfrak{m}\subset R/(p)$ are such that $R/\mathfrak{m}$ contains a copy of $\overline{\Bbb{F}_p((t))}$.