"Bougie" and "chandelle" both mean "candle" in French. What's the difference?
None already explain the technical and etymological difference.
The main difference is that in European French, or at least in France, chandelle is mostly used nowadays in set expressions, some of them already mentioned earlier in comments:
- brûler la chandelle par les deux bouts: to burn the candle at both ends, to do things in excess, to waste
- voir trente-six chandelles: to see stars, to be knocked out
- dîner aux chandelles: romantic dinner
- tenir la chandelle: "to hold the candle", to be a third wheel
- devoir une fière chandelle: to owe a great deal to someone.
- le jeu n'en veut pas la chandelle: something isn't worth the effort
- économies de bouts de chandelle: penny-pinching savings
As you can see, there are plenty of such idioms so the word itself is well alive, despite being almost unused alone. In the few cases where it is, the meaning is figurative:
- monter en chandelle/descendre en chandelle: to zoom (plane)
- faire une chandelle: to lob (e.g. tennis) or to throw a ball vertically (soccer and other sports), up and under, infield fly, pop fly, pop up, moon ball
- faire la chandelle: shoulderstand
- mettre une voiture sur des chandelles: to use jack-stands
The derived words chandelier/candelabre are also mostly used for antiques and substituted by bougeoir for current candle holders.
There is also a third word, cierge, specialized for candles to be lit in a church.
Canadian French kept chandelle as its main word for candle and mostly reserves bougies for spark plugs. That means in Québec, you would more likely hear souffler ses chandelles while in France, that would be souffler ses bougies. Both expressions are of course understood everywhere.
If we look at modern dictionaries chandelle and bougie are indeed sometimes given as synonyms and used as such by lots of people.
Nevertheless the use of the word varies according to French speaking countries. The BDLQ (Québec Board of the French Language) confirms what @gdupras says in their comment:
si en Europe chandelle est peu utilisé, au Québec, c’est un mot encore bien vivant, que l’on trouve tant dans l’étiquetage que dans les dictionnaires québécois comme synonyme de bougie. Ainsi, on ne saurait en déconseiller l’emploi1.
But strictly speaking bougies and chandelles are different objects, even if their purpose is the same. If you have, at home, or at the restaurant, a dîner aux chandelles you are not likely to have dinner lit by chandelles but by bougies.
Chandelles (candles) used to be made of tallow (suif in French) and sometimes (rarely) from wax. Tallow doesn't burn well and smells bad when hot. Candles made of wax became more widespread in France from the 14th century onwards because they started to import great quantities of wax from the town of Bougie (nowadays Béjaïa) in Algeria. And that's when the word bougie began to be used for wax candles in France.
In Britain wax also gradually replaced tallow in candle making but English didn't change the name as French did.
So strictly speaking chandelle is a tallow candle, and bougie is a (wax) candle.
Note that candle and chandelle both originate in the Latin candela.
1 if in Europe chandelle is little used, in Québec it is a word still very much alive that is found both in labelling and in Québec dictionaries as a synonym of bougie. Thus we do not discourage its use.