I am confused - isn't that what BRGs are for? Find a cheaper rate - go back to hotel - say hey here is cheaper rate? What am I missing?
The fundamental reason for BRG's is so that you book directly with the hotel chain and maintain loyalty with them. Most hotel scheme's do not earn points on OTA's or credits for the nights, as they have to pay commission to them. All the reasons trippin says also form part of the why from their side as well. However, you must follow the rules otherwise it is a waste of time.
From our point of view, it gives us the best of all worlds. We earn points, we get credit AND we get it at 20% cheaper than anyone else. You get all the benefits related to your status and it also means you can apply Suite Night Awards (if you have any).
For example, I recently stayed at Grosvenor House, Dubai. The base room was 1100AED + taxes a night prepaid. The only rate including breakfast was the flexi with breakfast at 1320AED+tax (approx). I found a rate of 867.17AED including breakfast on otel.com. My confirmed rate was 693.74AED+taxes including breakfast (867.17AED - 20%). I earned 5625 points and 3 nts credit for my 3 night stay. It was the equivalent of $292 per night including taxes & breakfast for the #3 rated hotel in Dubai. If I was staying 4 nights I found prices for 710AED (no breakfast) which would have come down to 568AED/nt. This was done 2 days beforehand. I have heard of people getting it under 500AED. 4for3 would have gotten it down to 990AED+taxes as a data point.
It is not easy to do it, however, because it is flexible, you book as early as you can and check it regularly.
The biggest difference to the 4for3 is that it is any room, any number of nights, up to 3 rooms, every hotel, no blackout dates (I BRG'ed Sheraton Time Square for New Years Eve) and almost every place can be done. But it takes work.