There is a difference between a "button-up" shirt and a "button-down" shirt. Learn it.
Forget it. I'll learn it for you.
A "button-down" shirt has little buttons to fasten down the collar. As opposed to; a tab that buttons under the tie, plastic or metal collar stays that get shoved in the corners of the collar and lost in the wash, or nothing at all except for a strong belief in the wearers own moral superiority. A "button-up" shirt has buttons on the front that hold it closed. As opposed to studs or, God forbid, a shirtfront. You would never have a button-down collar on a shirt that isn't a button-up, but you MAY have a button-up shirt that is not a button-down.
In other news, the "Today" show? A bunch of fucking idiots.