I also think that the rubber disperses the force of a horse stamping on the floor thus lessening a the chance of a horse putting its foot through the bottom so I would use mats but you need to lift them to stop water or worse getting trapped between the mat and the floor as this causes the wood to rot. I used to lift mine out in the winter as I didnt tend to do much then and it worked well and thats when the water seemed to get in as well.
I would use straw/shavings on the floor if I wasnt using mats but think mats are generally easier.
