They are pretty much interchangeable. I’d say ”fin” is closer to ”nice (looking)” and ”vacker” is closer to ”beautiful”.
Expanding on that, vacker (like beautiful) is usually about aesthetic appeal, while "fin" can be about high quality, a good person, something small and delicate, someone distinguished, etc. "Fin" could be translated to a number of different words including "nice", "fine", "excellent", "good", "delicate" etc. depending on context. Check here for a ton of examples (if the link doesn't work properly, just type fin in the search box there).
Absolutely right,there is really a little difference but people normally use them as synonyms
Isn't Trädgården also a bar in Sweden? Like from the song "Trädgården i fredag"