Customer service on twitter is a different ball game to phone and email support. For one your responses are open for the world to see. Customers can publicly slate you and you only have 140 characters to explain your side of the story. Some user’s reach is enormous, and they could easily spread their message about your perceived poor service to thousands, or even millions, of people very quickly.
Now I’ve spread the doom and gloom a little, let me see if I can help alleviate your worries.