Clay court tennis shoes give tennis players much more traction on clay courts than regular tennis shoes would. They are essential for any pro match played on clay and make a huge difference for the player.
Clay court tennis shoes are a very specific kind of tennis shoes that are designed only for use on clay tennis courts. They are one of the many different kinds of tennis shoes in the world, but are used only on one type of court rather than being versatile. The main difference between clay court tennis shoes and regular hard court shoes is that clay court shoes feature a full herringbone outsole, while hard court shoes have only a partial herringbone pattern. The herringbone is used to maximize the player's traction on the court so they can keep their footing during a match. Having maximum traction is important for tennis players on a clay court so they can slide naturally and be in control. They will have to slide when they are running to the ball and need to quickly stop.
There are many different kinds of tennis shoes for many different reasons. While there is no official tennis rule saying you have to wear a specific kind of shoe on a specific court, you should probably stick with what they were made for. For example, you can wear clay court shoes on a hard court, but you will not have very good grip on the court. Clay court shoes are not made to grip the court, while hard court shoes are. Also, if you wear clay court shoes on a hard court, they will not be as durable as they should. Another important aspect of playing tennis is comfort. Clay courts are much softer than any court, so clay court shoes do not need too much added comfort, but playing with them on hard surfaces will surely become uncomfortable.