Continuum robots deform continuously, compared to conventional robots composed of rigid links and joints, and require dedicated calibration methods. Indeed, calibration is an essential step to obtain high performance control, as it directly influences robot accuracy. In this paper, we investigate how model parameters influence both model accuracy and model-based closed-loop control accuracy of Concentric Tube Continuum Robots (CTCR). A fast, robust, and real-time implementation of the Cosserat rod model is first introduced. Then, a modelbased Jacobian control scheme is presented. A parsimonious calibration procedure focused on control accuracy is finally proposed to achieve submillimetric tracking errors along a 3D trajectory at velocity reaching 5 mm/s in complex scenarios including actuation constraints, obstacle avoidance, and external forces. Results are demonstrated both in simulation and on an experimental setup of a 3-tube CTCR.