.223 vs. Arrow/broadhead: I agree with Scorch
With the scenario you have stated in your thread, (all things being equal) they both will kill deer. However, I would choose the arrow with a sharp broadhead if it were me in this situation. As an avid deer hunter for the past 40+years, I have been fortunate to harvest many deer using both a rifle and a bow/arrow. Some comments on your question:
A couple years ago, I shot two deer (6 pt. and a doe) using my cousin's Remington Model Seven in .223 (Winchester 64 grain soft point bullet). The buck dropped on the spot and the doe did not go very far. This was the first time in all my years of hunting that I had used a .223 for deer. I wasn't that comfortable before the hunt but was impressed with the bullet's ability to take these deer. My cousin usually tries for neck shots with this gun....
But even now, I am really not that confident with this caliber for hunting whitetails. Reason? All of those shots were under 50 yards and I was fortunate to make perfect broadside hit on these deer. However, if they had been quartering away, were bigger size deer, behind any cover, a little farther out, etc, I don't think I would have taken the shot on them with this particular caliber. With that said, a .223 will kill a deer! Especially from 25 yards and closer, per the question presented.
However, with a bow and arrow and using a very sharp broadhead, you can take most any game here in North America (especially deer) with a well placed arrow. Especially the quartering away shots as it kills by hemmoraging and not by shock, etc. There is no size whitetail in the world, that a sharp broadhead can't put down, (if you do do your part and make the proper shot placement). I have experienced that that many times with my Mathews bow and a sharp (Rage 2-blade) and/or Muzzy broadhead. At 25 yards and closer, I would choose the arrow. Besides, to me it is a lot more challenging and more exciting when you harvest a whitetail by bow, (just my opinion). With that said, you need to practice a lot with your archery/hunting equipment before the hunt (to be fair to your game you are hunting). I love to hunt with both my rifle and archery equipment. However, when bowhunting season ends, I will pull out my trusty 6.5x55 for whitetails and leave the .223 to others for deer hunting. .