In the other answer linked above, it is said that the em-dash is the appropriate character to use for parenthetical text, which I too believe to be the case. On the linked pages about the Unicode characters, the horizontal bar has the index entries "quotation dash, HORIZONTAL BAR, BAR, HORIZONTAL dash, quotation" and the comment "quotation dash; long dash introducing quoted text". The comment for the em-dash is similar, but it doesn't have any reference to quotation in its index entries.
The index entries and (lack of) comment for the figure dash give no clues, but based on the name I'd guess that it's intended to be used with numbers, e.g. when giving ranges. But again, in the post linked above it is said that the en-dash is the correct dash to use for this purpose.
If anyone can answer, it would be interesting to know whether the shortcuts for unusual dash characters on Mac, Windows and Linux mightn't be creating figure dashes and horizontal bars as opposed to en- and em-dashes, which look very similar. Personally I'm using the X Windows shortcut
Ctrl-Shift-u plus the Unicode number, so they're all equally convenient for me—hence my desire to know what they're all for!