you can see more from the booth, and relay exact formation, down and distance tendency, etc. On the field you have direct contact with the players. You can give them reminders, encouragement, and get direct feed back from the field. In the old school it was one channel with all the booth coaches on the head coach's headset. Now I imagine multiple channels. Offense, Defense, Special Teams, as well as all on one. There is constant commentary from and to the both. What are they in? Is that an over/under? When they motion, they like to go weak side. Lets go nickel, lets go corner blitz.
I don't know how they are handling the booth and field right now, but its working. Possibly Miller in the booth and Phelps on the field. In the booth you need a coach who has seen it all. Someone who knows tendency and can stay a play or two ahead of the offense. On the field you need someone who can encourage, give players reminders, relay diagrams of what the both is saying. Instruction on how to play the option better. This individual must be a great communicator. Certainly the OC and possibly the QB Coach on the field are in constant contact. Same as the Defense. The head coach can but seldom does listen or comment. If he has an idea, he will run it through the booth and back to the field. Think of Hank Stram as he asked for them to check with the booth to see what 65 toss power trap looked like.
As the head coach, you are playing the game one, two steps ahead. Maybe making a call for a play, maybe a blitz, but most importantly you are in charge of strategy, the game plan, the game. Claeys, or Kill can get any information the booth has, but many times the game dictates that they have more important things to do.