Since the question

I am hoping to solicit examples of other counting problems of this type, particularly ones where the labeled count is known exactly but little is known about the unlabeled count.

does not make "exactly" a *sine qua non*, the following is a relevant example: **counting planar graphs**. It is still essentially an

open problem to find a formula, even an asymptotic one, for the total number of all unlabelled planar graphs on $n$ vertices.

(By the way, I think that counting unlabelled planar graphs is arguably the most intuitive combinatorial problem of all; after all, we are here speaking of drawing unlabelled maps, what could be more intuitive than that?)

In contrast, the corresponding problem for **labeled** planar graphs is not easy but much better understood, especially thanks to:

Omer Giménez, Marc Noy, *Asymptotic enumeration and limit laws of planar graphs*.

Journal of the American Mathematical Society 22, No. 2, 309-329 (2009).

it was (inter alia) proved for the first time that if $g_n$ denotes the number of **labelled** planar undirected graphs on vertex set $n$, then there exist two constants $\gamma,g>0$ and $g$, which are explicitly known (though not as elementary functions: there are *inverses* of elementary functions involved; but they can routinely be computed to arbitrary numerical precision), such that

${\Large\lim_{n\to\infty}\frac{g_n}{ {}\quad g\cdot n^{-\frac72}\cdot\gamma^n\cdot n!\ \quad}}\quad =\quad 1$. ${}\hspace{100pt}$ **(known)**

As far as I know, this is the best formula for the number of **labelled** planar graphs. (There seems to be no hope to ever find an explicit formula in elementary functions, or even a polynomial algorithm to arrive at the number $g_n$.)

In contrast, and this makes it relevant to the *question*, if $u_n$ denotes the number of **unlabeled** planar graphs on $n$, then all that is known is that

$\lim_{n\to\infty} (u_n)^{1/n}$

exists (and there is an explicit lower bound for this limit); this gives *far* less information than the result of Giménez and Noy. In particular, it is **not known** (and this is relevant to the OP's question) whether there exists a rational number $q>0$ and constants $u,\xi>0$ such that

${\Large\lim_{n\to\infty}\frac{u_n}{ {}\quad u\cdot n^{-q}\cdot\xi^n\cdot n!\ \quad}}\quad \overset{?}{=}\quad 1$. ${}\hspace{100pt}$ **(unknown)**