I'm going to answer this myself.
I've determined that there is a difference. "git fetch --tags" might bring in all the tags, but it doesn't bring in any new commits!
Turns out one has to do this to be totally "up to date", i.e. replicated a "git pull" without the merge:
$ git fetch --tags
$ git fetch
This is a shame, because it's twice as slow. If only "git fetch" had an option to do what it normally does and bring in all the tags.