I hope this question is not too soft for MO.
The Wikipedia says about finitism that it is an extreme form of constructivism. See http://en.wikipedia.org/wiki/Finitism. I doubt that this is correct.
As I understand, there were different approaches to solve the crisis of the foundation of mathematics. One was constructivism, where there must be a witness of an object. Finitism was another approach were one (Hilbert) tried to give existing mathematics a foundation in finitism. But it was not a rejection of certain mathematical methods, just finding a foundation. Finally, the third approach was just adapting the logics, which led to ZFC and type-theory.
Hilbert opposed the intuitionism of Brouwer. So, it is a little bit strange to count them to the same family.
In more modern finitism, related to reverse mathematics, one tries to prove that a finitism result obtained by infinitism methods, has a finitism proof. This has many successes and it has been shown that this is at least true for large parts of mathematics. Again, this is not a rejection of infinitism methods, but just an attempt to find a better foundation. While, constructivism is really a rejection of certain arguments.
So, based on above arguments, I believe the statement in the Wikipedia is totally wrong.
Maybe someone has more historical knowledge than me? If I am right, I can try to correct the article.
Edit: Thanks for the answers. I agree with Mike Schulman that both can mean a variety of things. I do think that the article in the Wikipedia needs some rewriting. It might be the case that finitism is more strict, however, I think it is not a subset of constructivism by definition (after lots of reasoning, one might conclude that).