This paper studies autonomous underwater vehicles (AUV)-assisted underwater acoustic sensor networks (UASNs), where AUVs act as mobile sensor nodes to collect information from areas not accessible to static nodes and then relay data via static nodes. Due to the difficulty of obtaining the accurate global information of all the static nodes, we propose a novel packet scheduling scheme by utilizing local information obtained by AUVs. In the proposed scheme, the localization of static nodes stage and the topology construction stage are carried out beforehand to obtain the local information, based on which the transmission scheduling stage is implemented. Furthermore, in the transmission scheduling stage, we design a value of information (VoI)-based packet transmission scheduling (VBPS) strategy to avoid packet collision. Specifically, we introduce a performance metric, i.e., VoI, to measure the importance of data packets with different levels of urgency. Then, we formulate a combinatorial optimization problem to maximize VoI taking packet collision avoidance into consideration. A low-complexity distributed search algorithm is proposed to solve the problem, which exploits the spatial-temporal reuse to establish data packet collision constraints and then determines the next-hop node and data transmission time for AUVs. Extensive simulations under various scenarios are carried out to evaluate the performance of the proposed algorithm.