A successful space launch mainly depends on the start-up process and variable operating conditions of rocket propulsion system. With the development of motor and battery technology, new electric pump is gradually used, in place of the traditional turbine pump, in rocket propulsion system. However, it is still unclear about the dynamic characteristics of the two processes in the two schemes (electric pump scheme and turbine pump scheme). In this research, simulation models of the basic components and overall system of an expander cycle hydrogen-oxygen engine were established on the basis of turbine pump and electric pump respectively. In the simulation models, the Centralized Parameter Method and the Stepwise Parameter Method are used for different components. The research results show that the power of hydrogen pump in the electric pump scheme is 25.5% lower than that in the turbine pump scheme, with speed and torque respectively 10.5% and 16.6% lower than those in the latter scheme. In the start-up process of the electric pump scheme, the flow of hydrogen pump underwent an overshoot of 3.7%, and the settling times of hydrogen and oxygen flows were both more than 40% shorter than those in the turbine pump scheme. Under variable operating conditions of turbine pump scheme, the speed of pump experienced an overshoot of about 20%, and the settling times of hydrogen and oxygen flows were 37% and 21% longer than those in the electric pump scheme. These findings are helpful to analyze the said two processes of expander cycle hydrogen-oxygen engine and further understand the difference in dynamic characteristics between the electric pump scheme and the turbine pump scheme.