OK, so if I go to the DOFF window and click on the department head tab for, say, Engineering, will the assignment list be different than if I go to my Engineering room and talk to the engineer there? I mean, I know there are some specialized missions available from visiting like empowering strange artifacts or building hypos, but is that the only difference? Or does one need to 'walk the ship' at every new zone when you're searching for that last chain mission?

The Department heads only give the standard missions, crafting missions and a few of the rare ones are only available in the interior.

However, there are only a few of the chain missions that are found on the interior, and you don't have to check your bridge every time you zone, the interior is treated as it's own zone, and the assignments in there are generated the same way as any other zone.

The missions from your Department Heads tab are the same as if you walked up to them on your bridge, except for as you already said, the specialty missions that are always there like fabricating stuff.

There used to be a difference, which was quite annoying, but it was a bug that was fixed quite a while ago. Hopefully, I haven't jinxed it now. <fingers crossed>