Both bath rugs and tub mats are designed to make the cold floor of the bathroom comfortable.
A bath rug is a decorative element of your bathroom, usually placed in front of the sink, by the door, or in the middle of the bathroom. It has a backing on it so it doesn’t slip and is stable even when water gets on the floor. It’s usually denser and thicker, with high absorbency and the ability to dry quickly.
A tub mat, on the other hand, usually matches your towel designs. It’s used to step out of the tub or shower and can be hung to dry on a rack.