Two reasons:
Geography: Finland isn't a part of the Scandinavian peninsula.
Language/Culture: The countries of Sweden, Denmark and Norway are traditionally Scandinavian, i.e. they speak North Germanic (Scandinavian) languages. The main language in Finland is Finnish, which is a Finnic language, which in turn belongs to the Uralic language family which is completely different language family than the one that the Germanic languages belong to (the Indo-European one).
Edit: I personally don't much like the definition "Scandinavia" in most contexts it's being used, and it certainly seems to be more used outside of northern Europe than within. "Norden" ("the Nordics") is the word most commonly used here in Sweden, and it includes Finland as well. I feel like it is an unnecessary distinction to talk about linguistical/cultural origins when speaking about the Nordic countries, because as it stands today, Finland and its culture, economy and politics is very similar to the rest of the Nordics, and political collaborations in northern Europe (such as the Nordic Passport Union) tend to be within the Nordic countries, not exclusively just the Scandinavian ones.
I think the main reason is that Sweden, Norway and Finland are very, very similar, while Finland, despite still being similar to those three, is not as similar.
Sweden, Norway and Denmark all share a common history - the Nordic mythology, the vikings, the Kalmar Union, owning Norway (Denmark owned it, Sweden was in a personal union with it, and now Norway owns it), they all have the same model of government (constitutional monarchy). Finland's only claim to this history is for a relatively brief period of time when Sweden ruled what is now Finland, and significant parts of it (the shared religious history, the Kalmar Union, the history of vikings, etc.) have nothing to do with Finland.
What's more, Finland was lost to Sweden before the idea of "Pan-Scandinavism" really developed (it really started as a political movement in the 1840's whereas Finland was lost in 1809). Although Pan-Scandinavism never resulted in the political union of these three countries, it did bring them closer together (for example, it lead to the Scandinavian Monetary Union). While all this was happening Finland (under Russian rule) was uninvolved.
So if you extend the term "Scandinavian" to include Finland you need to come up with a new term to refer to the "three brothers" and their common culture, language and history. For example, if "Scandinavian" refered to Finland as well you could not talk about the Scandinavian languages, or Pan-Scandinavism, as Finland would make those terms incorrect.
So there is enough difference between Finland and the other three to require two terms - one for only Denmark, Sweden and Norway, and one for those three + Finland (and others). And that is what we have: "Scandinavia" is just Denmark, Norway and Sweden (the three with the most in common) and "the Nordic countries" refers to the larger group.