The main reason is that Sweden, Norway and Denmark 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 Norse 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 (edit: Old Scandinavian has been dated to AD 600, or 1400 years ago. To me 500 years out of 1400 is "relatively brief", though obviously not insignificant. You may disagree.) 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.

I thought the Finns where a part of Scandinavia.

Also, what is the difference between Nordic and Scandinavia?

Follow up question. Why is Iceland sometimes not considered apart of Scandinavia? Is it just because it's too far away, despite the language, history, and culture it shares with Sweden, Denmark, and Norway?

Considering the nature of the cultural similarities between the Nordic/Scandinavian countries and especially Finland's place is not simple. There are many cultural similarities that tie Finland closer to Scandinavian than to the Baltic countries and Russia.

Finland still has Swedish as a second official language, there is a small but significant Swedish speaking population inside of Finland. Many of the Finnish nationalists of the early 1900s were bi-lingual or Swedish speakers. Furthermore the religious connection to Scandinavian Protestantism is very important as a cultural link. That is why although Finland 'feels' different to the Scandinavian visitors it also explains many of the cultural values that are also present in other Northern European countries.

Perhaps the most important factor however is the need of Finland to distance itself from its imperial heritage both of the Russians and of the Swedes. Independence and national identity in the early 1900s were both constructed by a Swedish speaking cultural elite. Lönrot's Kalevala (the national epic of Finland) and Sibelius' nationalist music were important factors in creating a cultural history and identity that until then had been denied to the Finns. ''The invention of tradition'' is certainly an applicable theory in application to Finland. As was the process of many subjugated nations emerging from declining imperial powers national identity and cultural identity was often defined in opposition to existing cultures and it would have detrimental for Finnish nationalism to emphasize their Swedish heritage too heavily. Instead nationalist Finns sought to distance themselves further.

However World War II saw a re-alignment with Scandinavia, and the post war governments shifted towards Nordic social-democratic model.

Ultimatley the cultural differences/similarities were present in the majority language of the Finnish people long before independence however it took the collapse of two different empires and a period of national awakening to solidify their identity, however there was an active and conscious distancing from Swedish and Russian identities from 1900s. However post-1945 and during the Cold War Finnish identity has re-align more with a Scandinavian image (social-democracy, middle-way economics, religious and cultural tolerance, political stability, welfare system) however many of the essential old cultural differences, and those created in the tide of nationalism, are now simply too prevalent to dismiss.