In mobile wireless sensor networks (MWSNs), each sensor has the ability not only to sense and transmit data but also to move to some specific location. Because the movement of sensors consumes much more power than that in sensing and communication, the problem of scheduling mobile sensors to cover all targets and maintain network connectivity such that the total movement distance of mobile sensors is minimized has received a great deal of attention. However, in reality, due to a limited budget or numerous targets, mobile sensors may be not enough to cover all targets or form a connected network. Therefore, targets must be weighted by their importance. The more important a target, the higher the weight of the target. A more general problem for target coverage and network connectivity, termed the Maximum Weighted Target Coverage and Sensor Connectivity with Limited Mobile Sensors (MWTCSCLMS) problem, is studied. In this paper, an approximation algorithm, termed the weighted-maximum-coverage-based algorithm (WMCBA), is proposed for the subproblem of the MWTCSCLMS problem. Based on the WMCBA, the Steiner-tree-based algorithm (STBA) is proposed for the MWTCSCLMS problem. Simulation results demonstrate that the STBA provides better performance than the other methods.