Both words have origins meaning a gathering of flowers — one from Greek and one from Latin. Both appear to have the same definition. When should I use one rather than the other?
Florilegium is simply a Late Latin calque (a literal translation) of the Greek anthology. L flōs/flōris = Gk ἄνθος (anthos) "flower, blossom"; LL legium = Gk λόγος (logos), "subject matter".
In English we only use the Greek-derived term.