My rubber snaffles are regular snaffles covered with black rubber. Obviously they are much softer than a regular snaffle due to the rubber, but they are stiff because there is a regular metal snaffle at the base. The happy mouth bits have more flex that a rubber because they don't have the metal underneath.
The rubber mullen mouth snaffles that I have do not have any metal inside, so they are very flexible. They are, however, thicker than the happy mouth mullen. I have ridden in both and they are both pretty flexible bits. I want to say the black is probably more flexible, but it really could be a draw.
The happy mouth mullen mouth is not as rigid as the snaffle happy mouth bits, if that's helpful.