What is the difference between the 8mm one and the 3/8'' one? Is it that the thicker one offers more protection?

So why you may want to choose the thicker Premium Mat is due to a few things. If you are matching up to another floor, you may want to use the thicker mat as not to have to add a transition piece. Another reason to go with a thicker mat is to help reduce noise transmission between floors. The thicker the mat, the more sound transmission it can mitigate. The last reason a person would go with the thicker mat is to add some protection for their subfloor, especially if the subfloor is plywood. The thicker mat will help to absorb the shock better from dropping weights and will diffuse it more to prevent damage.