Found this popped up on the right side panel, and thought I'd weigh in.
I keep my personal database alphabetically by latin name, but I also have a 'cheat-sheet' cross-index listing the common name alphabetically followed by latin. I try to not use the cross-index unless I have a brain freeze -- but I've also been known to simply paw through the pages looking for the correct photo when both the latin and common name are dancing on the tip of my tongue...
I usually use the common name when talking with others, as they are more likely to know what I am talking about. I am also not totally confident in my ability to pronounce the latin names correctly. And I don't want to appear stuffy.
On garden web sites, I use latin name with cultivar if known, with the common name in parentheses, for clarity and to accommodate both teams.
At a nursery, I go armed with a specific list down to the cultivar, but also have the common name in the event the nursery person is not well-versed in latin. I've had a lot of "huh...??" moments quickly resolved by throwing out the common name.