Are there post-quantum cryptosystems with a gap between classical and quantum security?
Here's an example where the best known quantum attack is, in a sense, just "halfway" between the best known classical attack on one side, and a complete break on the other:
Inverting a cryptographic group action such as CSIDH.

Let $G$ be a (finite) commutative group $G$ acting on a set $X$, i.e., we consider a map
$$ \ast\colon\; G\times X\to X$$
that is compatible with the group structure of $G$ in the sense that $1\ast x=x$ and $(g\cdot h)\ast x=g\ast(h\ast x)$.

The problem to be solved is analogous to the discrete-logarithm problem:


  Given two elements $x,y\in X$, find $g\in G$ with $g\ast x=y$, assuming it exists.




Classically, the best known attack is a meet-in-the-middle approach à la baby-step giant-step, where the group $G$ is split into two subsets $U,V\subseteq G$ such that $G=U\cdot V$, and one looks for a collision between the two sets $U\ast x$ and $V^{-1}\ast y$. Indeed, when $u\ast x=v^{-1}\ast y$, then $(u\cdot v)\ast x=y$. This takes time and space $O(\!\sqrt{\lvert G\rvert})$, which is exponential in $\log{\lvert G\rvert}$.
(In reality, there are better time-space tradeoffs than this simplistic approach.)

Quantumly however, this problem can be attacked using Kuperberg's algorithm [1,2] for the abelian hidden shift problem, and this takes subexponential (but superpolynomial) time in the group size. More concretely, the asymptotic cost is
$$ {\exp}\Big(\!\sqrt{\log{\lvert G\rvert}}+o(1)\Big) \text, $$
which can be roughly summarized as "taking a square root in the exponent".
(For comparison, the number field sieve for integer factorization is also subexponential, but with a cube root.)