Here's the long and short of it: in the current parlance, while many chignons are buns, few buns are chignons. Buns can be anywhere on your head, and you can wear lots of them at once. You can have them looped around your ears (see: Princess Leia), stuck up on your crown like Sailor Moon's little meatball odango buns, and worn messy, small, whatever.
Chignons, on the other hand, have traditionally been worn low on the head. They're also considered more formal, and you only wear one, usually large and containing the entire length of your hair.
The other notable difference? Buns are always wrapped around themselves, whether twisted around a center or braided. Chignons, however, aren't necessarily so. They may simply be tucked under or can be built around a central pin, tiara, or other hair accessory.
So while "chignon" technically just means a low bun, the word now carries connotations of formality and vintage style, as well as being used to describe updos that aren't buns. So there you have it; just remember never to try and argue about the word with a Francophile.