I worked in the hotel industry and I can tell you that there needs to be parity between 3rd party and direct booking. There is software that inspects it. Sometimes will you find a cheaper rate in either channel, yes, because not everyone follows the rules.
The one exception is loyalty programs. You will always get the best rate booking direct as a member of the loyalty program. It will be anywhere from 5-15% cheaper. You may be able to stack that with something like a special promotion (AARP for example).