I usually don't like filenames with extensions. The file type is rarely worth the user's notice, and it obscures the much more important filename with an ugly irrelevance. But the other day I didn't have extensions, and missed them. I was sorting through files on an old machine, and many were of types I hadn't seen in years. This was on a Mac, so the types were indicated by icons. And I had trouble telling what the types were.
Icons, as a rule, are easy to recognize when they're familiar, but inscrutable when they're not. (The early Mac UI designers had a saying: "a word is worth a thousand pictures", to a naive user.) This applies to icons for file types too. It's easier to distinguish unfamiliar types by name than by picture, because you don't know what the pictures mean yet.
This isn't really an advantage of extensions - it's just an advantage of text over icons. Of course there is a textual filetype column in the Finder, but it's too far away from the filename, and often too verbose, to be easy to read. ("Application program"? "ClarisWorks text document"?) Back when I used that machine, I hadn't needed that column, so I had turned it off in most folders, so it wasn't there to help me. But it wouldn't have been as helpful as extensions if it had been there. Extensions have three advantages: they're terse, they're easy to learn, and they're close to where you're already looking.
Okay, four advantages: they're also easy to change.
But so ugly.