In terms of the Community Feed, “0 milliseconds ago” typically appears when something happened at the same time you loaded the page—often because you loaded the page. For example, you receive the Jelly-Asker award at the very moment you submit your first question, so it should appear in the feed as having happened 0 milliseconds ago right after clicking “Ask Fluther” for the first time.
“Just a moment ago,” on the other hand, appears when something happened just before you loaded the page, but not simultaneously with you loading the page or because you loaded the page. So if someone gives a Great Answer or a Great Question just as you are clicking on a question, it should show up as having happened just a moment ago.
Or at least, this is what I have deduced from my experience with the Community Feed. Someone more familiar with the actual site code may be able to tell us more.