I feel that I should already know the answer to this, but it never sits quite right in my head. Let's look at something relatively concrete.

One way you can look at the moduli of hyperelliptic curves is by viewing them as double covers of a $(2g+2)$-marked $\mathbb{P}^1$. Naively, this seems fine; if we look at the dimension of these two spaces, they certainly agree. There is likely an issue regarding a choice of hyperelliptic involution, and an ordering of the marked points, but that's not really my concern here.

My concern is more the following. Which is the correct moduli space to look at?

On the one hand, we have simply the moduli space $\overline{\mathcal{M}}_{0, 2g+2}$ of genus 0, $(2g+2)$-marked points. One can by hand then build a double cover of any given curve of this form to obtain a hyperelliptic curve of genus $g$.

On the other hand, we have the moduli space $\overline{\mathcal{M}}_{0,2g+2}^{\mathbb{Z}/2}$ (for lack of better notation) consisting of those orbifold genus zero curves with $(2g+2)$ marked points each of which has $\mathbb{Z}/2$-isotropy. This definitely yields a map to the moduli space of hyperelliptic curves by taking the pullback via the diagram $$ \begin{align} & pt \\ & \downarrow \\ U \to & B\mathbb{Z}/2 \end{align} $$ where $U$ is any family of curves in $\overline{\mathcal{M}}_{0,2g+2}^{\mathbb{Z}/2}$.

This secondary approach is certainly more natural. As described, it comes with a natural map to the moduli space of marked hyperelliptic curves, as well as a natural map (by taking the coarse moduli space) to $\overline{\mathcal{M}}_{0,2g+2}$

I suppose then the main thrust of the question is: What is the difference (in terms of deformations, bundles, etc.) between the moduli spaces $\overline{\mathcal{M}}_{0,n}^{\mathbb{Z}/r}$ and $\overline{\mathcal{M}}_{0,n}$ where $r$ can be any integer? And what happens for higher genus? For example, I know that the tangent space at a smooth curve $(\Sigma, x_1, \ldots, x_n)$ in the moduli space $\overline{\mathcal{M}}_{0,n}$ is given by $$ H^1\big(\Sigma, T_\Sigma(-x_1 - \cdots - x_n)\big) $$

Side notes: I'm more interested in the case that $r = 2$, but I feel that I should ask more generally. Also, if the isotropy varies from marked point to marked point, then I would expect the issue to be more complicated, so I'm restricting to this simpler case. However, if there is no real difficulty in extending it, then by all means, do tell.