Matthews Public Library, first of all, is a gorgeous structure. It reminds me of the main Graduate Library at UNC-Chapel Hill (where I went to school...Go Tarheels!). The pillars and the circle drive and the lovely landscaping make it a great place to visit just for the ambience.
The Mecklenburg Library system is fabulous for many of the reasons that Diane D. describes in her review. I love all the online features and the ability for each branch to borrow from others.
One thing I've noticed to be particularly great about the Matthews branch is that they have so many special events. There's stuff for kids, like story hour and writing classes, but there are also interesting book signings, craft lessons, and other events for adults. I check their website out pretty often to see what is going to be happening, and I try to make as many events as I can. I'm particularly partial to the knitting and crocheting meetings. You can bring your projects and sit and chat with others who are working, and if you don't know what you're doing, it's a great place to get advice.