Not sure what you mean with unrelated exactly. They speak a language from a different language family (Finno-Ugric instead of Germanic) which I guess is the main difference. Though there exist noticeable minorities of Swedish speaking Finns along the southern and western coasts of Finland as well, dating back to the Middle Ages.

They don't really look any different from other Nordic people, culturally they are similar, and they are all Protestant Christians*, and Finland after the Nordic crusades in the 13th century ended up being part of the Kingdom of Sweden for six hundred years. So there is a lot of shared history with Scandinavian people too.

*The countries.

Well, the biggest reason I would attribute to Finland being Nordic but not Scandinavian is that Scandinavian is primarily about language (so you are correct in saying Scandinavian and Iceland as Icelandic isn't really Scandinavian) whereas Nordic is more cultural/political. Finland was ruled by Sweden for the majority of the past millennium.

Second, the term "nordic" is primarily Franco-German oriented. Nordic corresponds roughly to the North, as the Nordic countries are in Northern Europe, particularly the North Sea/Baltic Sea. Russia was always seen as the East/Eastern Europe. The Baltic countries for a long time were either controlled by Russia or Prussia, so not really "North" enough to count. And remember, that sovereign Denmark (as opposed to Norway which was controlled by Denmark) for the majority of it's existence included Skåne, which is now part of Sweden, so it had land on the "other side" of the Baltic sea.