You're a gem, Amanda! Quel joyau de question!

There you have it in a nutshell (mind you, I could have said "quel bijou!"). A joailler fits gemstones (joyaux) into jewellery (bijoux), which need not have gemstones. So there is an element of costume/precious jewellery, but it's not confined to that since in principle "joaillerie" has gemstones while a "bijou" could be simply a (worked) ornamental metal object (gold band, silver necklace, etc.), and not necessarily even precious metal, or metal at all, for that matter.

Bref:
joaillerie = ornamental objects (preferably of precious metals) fitted with precious stones;
bijouterie = ornamental objects, preferably of precious metals, with or without precious stones (?)

And as others have said, we say "jewellery" for both. Things are further complicated in French by the fact that an "orfèvre" (orfèvre-joaillier or orfèvre-bijoutier, as opposed to the orfèvre who makes silver tableware) can make bijoux and set joyaux into them!

While a Frenchman, in the good old days of wind-up watches with jewels for bearings, took his watch to his high-street "horloger", we took ours to the nearest jeweller, so in English the term covers yet another field. "Only a jeweller can sell an Edo*".