Online and offline channels compete on product price and service simultaneously, which is significantly influenced by supply chain structure. The differential game method is implemented to study the competition between online and offline channels in three types of supply chain structures, including centralized structure, decentralized structure, and revenue-sharing structure. By solving the differential game model, we obtain online and offline retail price, promotion effort, and logistics effort strategies under different supply chain structures. It is found that the decentralized structure increases offline channel's retail price and promotion effort, and online channel's retail price, while reduces online channel's promotion effort and logistics effort. The revenue-sharing structure increases offline channel's retail price and promotion effort, online channel's retail price and logistics effort, and reduces online channel's promotion effort. The decentralized and revenue-sharing structures enable the e-tailer to make a smaller promotion effort than offline promotion effort and to set a higher retail price. When the marginal contribution of logistics effort is sufficiently large, the existence of logistics effort always leads to a higher goodwill for the online channel than offline channel. Numerical experiments show that the revenue-sharing structure is most advantageous for the offline channel, while centralized structure is most advantageous for the online channel. Whether logistics effort is beneficial to the online channel depends on its marginal contribution to the goodwill of online products.