Back when I was in Russia I'd say it'd be Suzdal, famous for the density of churches and other traditional architecture; or Tarusa, known for that one song that everyone seems to know a couple of lines from, Gorodok (here is a random rendition I found just now), both with just under 10k pop according to the wiki. And, as a bit of a stretch since it's not a town and most people would call it Solovki, Solovetsky settlement, famous for being a prison, with about 800 people. Also Oymyakon with under 600 people, the coldest settlement on earth if you're into that sort of thing.

Now in Georgia, I'd say Borjomi with just over 10k pop famous for its water, and Bakuriani (just over 1800 people) for its water and the ski resort. Again a bit of a stretch, but I guess everyone in Georgia at least also knows the ski resort of Gudauri at just under 100 people, as well as mountain resorts of Gomismta and Bakhmaro, both with no permanent residents due to the rough winters. Geography nerds will also be familiar with Ushguli, (arguably) the highest inhabited settlement in Europe, population 220.