My sensei's dojo uses cheerleading floor like the ones that can be found at US Gymmats
. They are two inches thick with a thin carpet bonded to the top. They also come with velcro that you put down to cover the seams (the velcro sticks directly to the carpet). They really are quite nice. There isn't a sprung floor under it, just a 2x4 and plywood sub-floor to bring the floor up to the correct height.
The dojo I teach in has the tire type of sprung floor with Swain mats. All in all I prefer the cheerleading mats on the sub-floor. Firm without being hard and cushions without being bouncy. I also like the fact that the seams are covered...I've twisted a couple toes between the Swain mats). The tire sprung floor is also uneven. There are peaks and valleys in the floor and some definite "soft spots" here and there. The cheer flooring is very consistent.