As for what goes on: it's largely only necessarily for "wholesale" trading, or trading in large size.
There's a bid and offer price for every derivative or asset, not just one price.
Taking SPX options as an example, screens may be 40c wide on an option for only 200x size. Often times you need to buy or sell up to 5000x options, in which case the floor can provide much more size at similar or often times better prices than screens. This is especially true for complex orders.