We all have a story of a company with such awful customer service that we swore we would never return and told all of our friends to never go back either. When companies handle just one situation badly, it can have drastic effects on future business for a long time. The same holds true when you're an independent contractor.
As an IT contractor, you are running a business and customer service should be as important to you as it is to any other company. The way you interact with clients, project managers, your team, and any of the client's employees will play a role in where you work in the future. A negative review by any of these people could affect whether or not the client extends your contract, your agency calls you for a new gig, or a colleague recommends you to a recruiter down the road.
A common downfall in customer service is conflict resolution. You're challenged to ensure problems are solved, the blow to your business is minimal, and the client leaves happy. This is a hard balance. Fortunately, this video from Executive Leadership Training can help with great advice for resolving conflict around the office.