I'm currently a senior research software engineer at a large company and am being offered a "senior staff engineer" position somewhere else. I am not sure if the new position's title conveys a sideways move or an advancement.
So, in your opinion, what is the external difference between these software engineer titles (in general and regardless of any particular company, if possible):
Honestly, I've found that trying to compare titles at different companies is an exercise in futility - not only are they often on different scales, but the same title might mean a completely different thing at one company than it would another.
For instance, some places have regular "Software Engineers" and then have "Senior Software Engineers"; some have "Associate Software Engineers" and then just "Software Engineers", et cetera.
Instead, look at what the job actually is - what kinds of responsibilities do you have, where would you fit into the hierarchy, how much does it pay (if you already have an offer in that regard), and so on.
If the 'engineer' title is regulated in the jurisdiction you work in there might be something to it, otherwise you'll probably just have to go by what the responsibilities are, the level of accountability you have for your decisions and the salary and benefits package.
More importantly you have to consider your goals. What are you trying to advance towards?