Gollum is useful for editing a local, Markdown-based wiki and is mostly compatible with Github's Wiki. Below are lessons learned about the differences between the two to avoid headaches when local edits in Gollum don't produce the desired effect in Github Wiki.
TL;DR In general, there are inconsistencies between the two and one must target Github Wiki's features even if they don't work in Gollum if you don't want to restrict yourself to a compatible subset of the two.
NOTE: This has been derived from a private, Github Enterprise instance rather than the public Github instance. In theory that should not affect things...
Github's Wiki documentation explains that both WikiLink syntax ([[Text]]) and Markdown syntax ([Text](link)) is supported.
This Github gist looks useful, though is several years old.
The WikiLinks syntax ([[Text|link]]) is supported in both Gollum and Github Wiki pages for internal pages. link should not have a .md suffix. External links use the Markdown syntax of [Text](link) and link should have a suffix.
Note that this is different than Github's Markdown rendering within a repository, where the Markdown link syntax is used.
One discrepancy between the two is where an internal link is used within a table and the vertical bar (|) must be escaped within the link, otherwise it will be misparsed as a table cell delimiter. That is, links must look like [[Text\|link]] inside of table cells.
NOTE: Gollum v5.3.0 will render Text\ as the link's text, while Github Wiki will render it as Text.
Gollum and Github Wiki do not render local, internal link anchors the same way. That is [[Text|#anchor]] works fine on Gollum, but creates a link to #anchor#anchor in Github Wiki. Explicitly referencing the current page (which makes the link more brittle as it is now page-specific) seems to work in both, like so [[Text|Current Page#anchor]].
Gollum and Github Wiki normalize local, internal link anchor names the same way.
Gollum will normalize Ubuntu 22.04 to #ubuntu-22-04 while Github Wiki will use #ubuntu-2204.