In contrast to /u/Gilad1's post: in late medieval Europe, public baths at hot springs were an important part of the medical economy. Indeed, bath house attendants were within the spectrum of medical practitioners! (doctors, barber-surgeons, executioners, etc). Twelfth-century nun Hildegard of Bingen encourage her readers to breathe in the vapor of hot springs (heated by the fires of purgatory, no less!) to absorb excess humors knocking their health off balance. Fifteenth-century jack of all trades Hans Folz spent several years traveling southern Germany, and eventually published a guide to the region's public baths. There were dedicated bath houses in cities as well. The operators of one Nuremberg bathhouse are quite notable in city records for their propensity to sue their neighbors.

They seem to have died out (or are legally abolished) in western Europe instead with the changes surrounding the Reformation--increasing public regulation of morality, growing awareness of how disease spread, syphilis epidemics, and the entrenchment of views that public baths were storehouses of sin and prostitution, not health and healing. The most famous case is 1546 England, with notable action happening in France clustered in the 1550s.

I don't know whether there is any continuity between the Roman tradition and the late medieval one (I mean, obviously men and women weren't enjoying lavish meals whilst naked in tubs in Nuremberg bathhouses when Nuremberg didn't yet exist), or how the medieval version might have developed in the early and high M.A. Virginia Smith's Clean: A History of Personal Hygiene and Purity might help you there.

Maybe I'm missing the point (not to mention probably breaking the rule against discussing the present day), but there are still numerous public saunas in Finnish cities. My own question would be: How does the Finnish sauna tradition diverge from medieval bathhouses elsewhere in Europe?

Disclaimer - not an expert on Europe post fall of the WRE (Western Roman Empire). However I know a lot about bath houses and how they were built / maintained so I can provide some speculative insight.

Basically bath houses were public structures built and maintained by the local elite. It was giving to the public to be recognized and win votes for local elections. When the western Roman Empire started disintegrating to non-Romanized peoples, the local elites had to worry about how to maintain their wealth / life with these foreign invaders. So due to this the use of bath houses likely fell out of favor since the Roman cultural traditions were not maintained by the ruling elite and the maintenance and building of public bath houses no longer functioned, causing the bath houses to fall into disrepair and not for use.

Again, I'm not an expert on this particular period, just what I think is plausible for Western Europe and wanted to give you some feedback!

That said, no idea if Byzantines discontinued the practice at some point, so can't help you there.