To which extent the adjunction $F\dashv N\_\omega$ generated by the $\omega$-nerve described at [$n$Lab - oriental](http://ncatlab.org/nlab/show/oriental) (obtained as a particular instance of the nerve-realization paradigm) is linked to the adjunction generated by the functor $O\_{[\Theta]}\colon \Theta\to \textbf{Str}\text{-}\omega\text{-}\mathbf{Cat}$ ([Joyal's $\Theta$-category](http://ncatlab.org/nlab/show/Theta+category)), 
$$\text{Lan}\_y(O\_{[\Theta]})\dashv N\_{[\Theta]},$$
where the functor $N\_{[\Theta]}\colon \textbf{Str}\text{-}\omega\text{-}\mathbf{Cat} \to [\Theta^{op}, \mathbf{Sets}]$ sends $C\in \textbf{Str}\text{-}\omega\text{-}\mathbf{Cat}$ to the presheaf $\textbf{Str}\text{-}\omega\text{-}\mathbf{Cat}(O(-), C)$? Is there any reference to learn about, and quote properly, affinities and differences between the two?



$\omega$-nerve versus $\Theta$-nerve
As you remark, the cellular and simplicial nerves arise from the functors $J \colon \Theta \longrightarrow \omega\text{-Cat}$ and $O \colon \Delta \longrightarrow \omega\text{-Cat}$ respectively, where $J$ is the full inclusion of Joyal's cell category and $O$ is Street's orientals functor. 


A significant difference between these two is that $J$ is *dense*, and hence the nerve functor $\omega\text{-Cat} \longrightarrow [\Theta^{op},\text{Set}]$ is *fully faithful*, whereas this is not true of $O$ and the simplicial nerve induced by the orientals. (For a reference, this comment is made in the introduction to Section 1 of Berger's '*A cellular nerve for higher categories*'). Note however that the modification of the simplicial nerve which lands in *stratified* simplicial sets is fully faithful (see Verity's '*Complicial sets...*', whose introduction contains an account of the history of this nerve).


The cellular nerve has these nice properties because it arises from general theory; I mean the *nerve theorem* (see for instance the Introduction and Theorem 1.10 of Berger, Melliès & Weber's '*Monads with arities...*'), which applies upon recognising the objects of $\Theta$ as the free $\omega$-categories on the canonical arities for the free $\omega$-category monad on globular sets.


Sadly, there is no such nice theoretical description of the orientals functor; the best we can give is an explicit construction using parity complexes or similar. 
Yet there is still another description of the simplicial nerve of a 2-category (see section 10 on nerves in Lack's '*A 2-categories companion*'). The composite functor $\Delta \longrightarrow \text{Cat} \longrightarrow 2\text{-Cat}\_{\text{nlax}}$, where $2\text{-Cat}\_{\text{nlax}}$ is the category of 2-categories and normal lax 2-functors, is dense and so gives a fully faithful nerve functor $2\text{-Cat}\_{\text{nlax}} \longrightarrow [\Delta^{op},\text{Set}]$ which agrees (up to duality) on $2\text{-Cat}$ with the nerve induced by the orientals. So we could hope that, if we had a category $\omega\text{-Cat}\_{\text{nlax}}$ whose morphisms are "normal lax $\omega$-functors", the composite functor $\Delta \longrightarrow \text{Cat} \longrightarrow \omega\text{-Cat}\_{\text{nlax}}$ would be dense and give a fully faithful nerve functor $\omega\text{-Cat}\_{\text{nlax}} \longrightarrow [\Delta^{op},\text{Set}]$ extending the simplicial nerve given by the orientals. This suggests the tentative theoretical description of the $n$-th oriental as the "normal lax $\omega$-functor classifier" of $[n]$. However, I do not know of any definition of normal lax $\omega$-functor, which would likely make use of the orientals anyway.


With regard to their affinities, I am not aware of any study in the literature. At the heart of the comparison is the module $\omega\text{-Cat}(J,O) \colon \Theta \nrightarrow \Delta$, corresponding to the functor $\Delta \longrightarrow [\Theta^{op},\text{Set}]$ that gives the cellular nerves of the orientals. One could hope that this module has a nice combinatorial description. Perhaps someone out there has considered this, maybe in the comparison of the models for higher categories based on weak complicial sets and cellular sets.


