A mat has a protective function. You put down a mat in order to protect the floor. You might have a soft mat in the bathroom to absorb drips of water when you get out of the bath or shower, a rough mat (made of coir or sisal) in the doorway of your house to pick up dust and dirt from your shoes, or a wipe-clean plastic mat to put under a baby's highchair in case any food or drink might fall. We also use mats in places other than floors. For example, if you want to protect the surface of a table, you would put a small mat under a drink or a plate. A rug is generally larger, and usually has a decorative function. You put down a rug to make a room more attractive or comfortable - it's like a small carpet.

When I think of a rug: I think of a house rug When I think of a mat: I think of a welcome mat when you dust your shoes off before going inside, a dog mat where the puppy sleeps or place mat on the dinner table. The difference is a mat is everyday use item sometimes made out of a coarse woven material (dog mat, place mat, welcome mat) and a rug is nice, pretty, a piece of furniture (like a nice living room rug) ... hope this makes sense.

Mats are small and not fluffy at all. Rugs are a lot bigger with fluffy fibers. Mats are usually only used in bathroom and toilets eg "bathroom mat", while rugs are usually used in the sitting rooms and dining rooms etc on the floor and couches.