It's hard to find good pictures, but a casual button down is usually cut pretty much the same all around the bottom. There will be a tad bit more fabric on the front and back than on the sides, but not much, like so :
A dress button down is cut so there is a lot more fabric in the front and the bottom than on the sides. It's so when you move around with the shirt tucked in, your waist moving (from such things as sitting and standing) won't untuck the shirt in the front or back. I hope you can see from this pic :
That one is an exaggerated dress button down. Not all are that exaggerated but hopefully you see my point that it looks stupid to wear the second shirt untucked. Many people still do it but I hear from women all the time how sloppy it looks and there was also an article in GQ about it not too long ago.