To answer your question:

Is there a graph class, listed in the above database of graph classes, where Hamiltonian Cycle and Hamiltonian Path problems have different complexity?

I think the answer is 'no' (unless there is a bug in my code). The list of all classes categorized by their complexity status is given for each of Ham-path and Ham-cycle:

https://www.graphclasses.org/classes/problem_Hamiltonian_path.html
https://www.graphclasses.org/classes/problem_Hamiltonian_cycle.html

I tested everything Linear or Polynomial vs all other categories (GI-complete, NP-Hard, NP-Complete, Unknown to ISGCI).

Surprisingly (or not?) there are no classes that show up with a polynomial mismatch except when it is compared to the `Unknown' classification. But what is furthermore somewhat surprising is that in all these cases, it is Ham-Cycle which is known polytime, while Ham-Path is the unknown one.

The classes I found were:

- 66 (biconvex) has linear HAM-cycle but Unknown to ISGCI HAM-path
- 67 (convex) has linear HAM-cycle but Unknown to ISGCI HAM-path
- 407 (($P_5$,claw)-free) has linear HAM-cycle but Unknown to ISGCI HAM-path
- 508 ((2$K_2$,claw)-free) has linear HAM-cycle but Unknown to ISGCI HAM-path
- 645 (equiv to biconvex) has linear HAM-cycle but Unknown to ISGCI HAM-path
- 1144 (claw-free locally connected) has linear HAM-cycle but Unknown to ISGCI HAM-path
- 1146 ($K_{1,4}$-free, locally connected, almost claw-free) has linear HAM-cycle but Unknown to ISGCI HAM-path
- 1234 (($P_6$,claw)-free) has linear HAM-cycle but Unknown to ISGCI HAM-path
- 644 (circular convex bipartite) has polynomial HAM-cycle but Unknown to ISGCI HAM-path
- 1058 (solid grid - you mentioned above) has polynomial HAM-cycle but Unknown to ISGCI HAM-path
- 1094 (locally connected and max deg 4) has polynomial HAM-cycle but Unknown to ISGCI HAM-path
- 1142 (2-connected $\cap$ linearly convex triangular grid graph) has polynomial HAM-cycle but Unknown to ISGCI HAM-path
- 1143 (locally connected $\cap$ triangular grid) has polynomial HAM-cycle but Unknown to ISGCI HAM-path
- 1197 (adjoint graphs) has polynomial HAM-cycle but Unknown to ISGCI HAM-path
- 1198 (quasi-adjoint graphs) has polynomial HAM-cycle but Unknown to ISGCI HAM-path
- 1199 (directed line graphs) has polynomial HAM-cycle but Unknown to ISGCI HAM-path
- 1201 (equiv to directed line) has polynomial HAM-cycle but Unknown to ISGCI HAM-path