Fairness among different users and energy utilization are key issues in the future communication network design. Robust max-min fairness resource allocation in sensing-based wideband cognitive radio with simultaneous wireless information and power transfer is studied when spectrum sensing and channel state information are imperfect. A worst-case throughput is maximized by jointly optimizing the sensing time, transmit power, and subchannel allocation under the worst-case channel state information error model, subject to constraints on energy harvesting, interference power, and transmit power. Two operation paradigms for cognitive radio are considered, namely, opportunistic spectrum access and sensing-based spectrum sharing. The formulated robust max-min fairness resource allocation problems are mixed-integer and nonconvex programming with infinite inequality constraints. An efficient one-dimensional search algorithm is designed based on the proposed transmit power and subchannel allocation scheme. Simulation results show that the secondary user under sensing-based spectrum sharing can obtain a performance gain compared with that under opportunistic spectrum access at the cost of implementation complexity. Design tradeoffs are identified and discussed.