Paper cards have been in existence probably since the first card was invented. It has always been the material of choice for the required properties of these things, which is flexibility, for easy shuffling, and portability.
Plastic cards, on the other hand, do not have similar disadvantages as the paper ones. These cards are very glossy making it easy to handle almost as if there is a cushion of air between the cards. Plastic cards are extremely flexible, snapping back to its original shape after being bent. These cards are washable, once soiled it can be cleaned with soap & water and they are waterproof. The disadvantage of plastic cards is that they do not have the texture of paper.