What advantage do the 3/4 inch (Extreme) have over the 1/2 inch (Mega)? In other words, what does that extra 1/4 inch of thickness get you?

Really there isn't an advantage of going with the 3/4" Extreme Mats over our 1/2" Mega Mats. They both have been designed for dropping heavy weights and are very shock absorbent. The only time we would highly suggest someone going with the 3/4" over the 1/2" is if they have plywood subfloor and if they are on a second floor. That extra bit of material helps to absorb more of the shock when dropping weight better, protecting your subfloor and also deadening more of the sound. If you have a concrete subfloor and are on the first floor, the 1/2" Mega Mats are plenty.