What is the difference between a National Merit Scholar and a National Merit Finalist? Is the later just missing the cut?

Yes, basically. Unless things have changed since I had the one and not the other. Finalists are the short list of candidates, and scholars are those who end up being chosen out of that short list.

EDIT. I seem to remember that to be a National Merit Scholar meant that a school (college) gave you a scholarship (a National Merit Scholarship) and you could only apply for it at one school, but it was a very long time ago and the memory is vague. I remember that the school I selected (and did not actually end up attending) did not award me the scholarship. But maybe that is baloney that my memory invented for me.