From a mat-management point of view on an HFT course - the big difference is that the Rangesports mat is W shaped and the AIM mat is Z shaped.
The handles on the RS are on the hinges in the middle so that they stay clean and you can grab one and then the other on the way up from the shot, like a concertina. No mud gets on your hands and then your gun.
The handles on the AIM are at the end, where they get in the mud. You can grab one end of the mat but you can't then get the other end to gather it up, 'cos it doesn't fold as easily, so it's a bit more of a phaff. There are rubber rings on the corners which make carrying it easier once it's folded. Some folk use elastic string to 'join' the handles together to ease grabbing the middle of the mat.
The AIM mat is big and comfy but my choice would be the RS. I think most of the ground we shoot on in HFT has as much springyness as a mat, so I don't necessarily see Vinnie's point