Mobile edge computing (MEC) is an emerging paradigm to offload computations from the cloud to the MEC servers in vehicular networks, aiming at better supporting computation-intensive services with requirements of low latency and real-time processing. In this work, we investigate a new service scenario of computation offloading and workload balancing in MEC-empowered vehicular networks, where the computational resources of MEC/cloud servers are cooperatively utilized. Then, we formulate a distributed task assignment (DTA) problem by considering heterogeneous computation resources, high mobility of vehicles and uneven distribution of workloads, targeting at optimizing task assignment among MEC/cloud servers and minimizing task completion time. We prove that the DTA is NP-hard. Further, we propose a multi-armed bandit learning algorithm called Utility-table based Learning. For workload balancing among MEC servers, a utility table is established to determine the optimal solution by online learning of real-time workload distribution, which is updated based on the feedback signal of task assignment. For optimal computation offloading, a theoretical bound is derived to determine the ratio of workload assigned to the cloud. Lastly, we build the simulation model and conduct an extensive experiment, which demonstrates the superiority of the proposed algorithm.