Knee braces are designed and used to protect a previous injury from further endangerment. Knee sleeves are designed to protect the knee from future injury or risk of damage. This protection is especially important for knees put under great daily pressure (running, jumping, weightlifting).
Knee sleeves also add a valuable compression element that increases blood flow and reduces pain, not only during but also after the performance. The reason this compression aspect is so important is that a compressed knee encourages blood flow through the blood vessels of the knee. Here is how I would draw it up on a chalkboard: compression + blood flow = better recovery. Simply put, using a knee sleeve results in less pain and swelling during and after a performance.
Knee compression sleeves are generally made from neoprene material and slide on over the knee. In simple terms, the idea behind the knee sleeve is to reduce pain. More specifically, the sleeve adds warmth, limits patella movement, and can increase proprioception (the capacity to feel the position of a joint in space as sensed by the central nervous system). In other words, the sleeve is more than a mechanical support mechanism for the joint but is also used to improve proprioception.