Let ${\mathcal C}$ be a simplicial category. Then there are the following two ways of constructing a simplicial set from ${\mathcal C}$:
Form the simplicial nerve $\text{N}_\Delta({\mathcal C}) := \text{Hom}_{\Delta\textsf{-Cat}}(\Delta^{\bullet},{\mathcal C})$, where $\Delta^{\bullet}$ is the cosimplicial simplicial category with $\Delta^n$ a simplicial category version of the $n$-simplex (Lurie, Def. 1.1.5.5)
View ${\mathcal C}$ as a category in $\textsf{sSet}$ with discrete space $X^0$ of objects, morphisms $X^1 := \bigsqcup_{x,y\in\text{Obj}({\mathcal C})}{\mathcal C}(x,y)$, and form the geometric realization of its internal nerve (a simplicial object in $\textsf{sSet}$) $$\widetilde{\text{N}}_\Delta({\mathcal C}) := \text{Real}\left(\ldots\substack{\longrightarrow\\[-1em] \longrightarrow \\[-1em] \longrightarrow\\[-1em]\longrightarrow}X^1\times_{X^0} X^1 \substack{\longrightarrow\\[-1em] \longrightarrow \\[-1em] \longrightarrow}X^1\rightrightarrows X^0\right)$$
How can these two constructions be related? I'd also be happy if someone could provide a good reference.
Here is some vague intuition on why there might be a relation: First, in the case of an ordinary category viewed as a discrete simplicial category, both constructions agree with the ordinary nerve construction (any simplicial set is the geometric realization of itself when considered as a discrete simplicial space). Also, looking at $2$-simplices, the two constructions are similar: In the simplicial nerve $\text{N}_\Delta({\mathcal C})$, a $2$-simplex is given by morphisms $\alpha: x\to y, \beta: y\to z, \gamma: x\to z$ together with a homotopy $H: \beta\alpha\to\gamma$, so $2$-simplices in $\text{N}_\Delta({\mathcal C})$ correspond to compositions $(\alpha,\beta,\gamma,H)$ of $1$-morphisms up to homotopy. For $\widetilde{\text{N}}_\Delta({\mathcal C})$, $2$-simplices coming from the $0$-simplices of $X^1\times_{X^0}X^1$ only account for strict compositions $(\alpha,\beta,\beta\alpha,\text{id})$ in ${\mathcal C}$, but this is compensated by the fact that the $1$-simplices of $\widetilde{\text{N}}_\Delta({\mathcal C})$ coming from the $0$-simplices of $X^1$ vary 'continuously' in $X^1$.