Tennis shoes offer numerous advantages over casual trainers. The main difference, is that tennis shoes are designed to cope with and support the player during the dynamic movements they go through when moving across the court.
Other types of shoes such as running shoes are constructed with a focus on forward movement and therefore will not be suitable for tennis. Tennis shoes take into account the way a tennis players foot moves so they are able to add the right support and comfort exactly where it is required e.g. improved ankle support, and extra heel cushioning.
Other differences are more visible like the toe guard which helps protect the shoe from toe drag. Shoes without a toe guard may easily show signs of wear and tear after a few sessions on court.
It should be noted that some of the best tennis shoe brands also manufacture casual trainers and footwear for other sports, however you must choose from the tennis range. e.g. Asics make some fantastic tennis shoes, but their casual trainers and running shoes will not have the features required for tennis.