The general rule of thumb is that a non-button shirt is considered to be more formal than it's counterpart, the button down collar. Both are considered appropriate for the work place. For me, if I am wearing a suit, I will make sure that I go with a more formal, non-button collar. If I am just going with a tie or a tie and sport coat/blazer, either option will work!
Another option is the white collar with different color shirt...wouldn't be my everyday selection, but a fun choice to rotate into your wardrobe.
Does all that make sense? Button down = less formal, no buttons = more formal. Next up on my dress for success tips...knots for your tie!