The short answer: probably not…The short answer: probably not, gameplay-wise…

Steam and GOG use different directory structures, so the installed/unpacked folders are not exactly the same.

On the other hand, I found only one "visible" difference so far since version 1.2.1 (from round about 2014, downloaded 2016) of the game between this old edition and a higher 1.2.x (1.2.6? probably 1.2.8): the teleportation item you can get seemed to work "longer" or throughout all the game i old versions. This is probaly due to the end being rather linear involving time travel. The item would have transported you back (if working at this point) to the "present time" with no way back into the time travel sequence of events.

Note that this is an AGS (Adventure Game Studio) game, making things much more complicated. AGS games need a runtime engine, which can be built-in on Windows (like in this case), is standalone on non-Windows, and can be played with a standalone engine of another version as is built-in even on windows.
So there is the game version (currently 1.2.9), the AGS version the game distribution package was built on, and the AGS runtime version you play it with. Additionally there can be differences between available versions depending on the platform (OS X, Linux, Windows) between GOG and Steam.

This can't be easily found out by other means than installing it from both GOG and Steam, and comparing the files, because they often changed their downloads in the past, it wouldn't even help you if I remembered all exact details from my own findings, see below.

I'm using Linux here, installed mine from an/the AGS game site in 2016. At this time there was a limited supply of non-matching native Linux AGS runtime versions available, so I started using the Windows version via Wine. After that, I found the game on Steam, installed from there, "merged" the 2 installations by hand to use the latest files and my saved games. On Steam was probably version 1.2.8 or something between 1.2.1 and 1.2.8 with that "teleportation difference" mentioned above being the only difference I noticed. When it appeared on GOG, I saw that it had "arrived" at game version 1.2.9, I repeated the "merging" an re-played it half-way so far, finding no noticeable difference.
BUT the Windows version on GOG: "setup_heroines_quest_the_herald_of_ragnarok_1.2.9_(67647).exe" was >443MB, slightly bigger that my old 1.2.1 download from elsewhere, the Linux version: "67882" only >159MB. Then, a few days later, another one for Linux appeared: "heroine_s_quest_the_herald_of_ragnarok_1_2_9_with_dlc_68122.sh" >443MB, no idea what the "DLC" is (probably [not] included PDF and picture files). The current one is: "heroine_s_quest_the_herald_of_ragnarok_1_2_9_2_68339.sh" >443MB. Due to my installing the newer versions simply over the old ones, I can't tell the differences exactly.

The Linux versions come with separate AGS runtimes in the game packages. Apart from this, there is no "real" difference between Linux and Windows versions.

Summary: I doubt that they differ in any other way than in "contained programming bugs".

Almost.

The Steam version connects to Steam to use Steam achievements, whereas the GOG version connects to GOG to use GOG achievements. Other than that, the game should be identical, and if and when we do an update, we put it on both sites at more-or-less he same time.