The variable speed pumped storage unit is an important means for adjusting the power fluctuation of a system. Rotor winding short-circuit and rotor eccentricity faults are common fault types. Both faults will generate harmonic circulations with similar characteristic frequency bands on the stator side. These lead to a difficulty in distinguishing between the two faults. In this paper, a fault diagnosis method based on fast Fourier transform-long short-term memory (FFT-LSTM) is proposed to distinguish between the rotor winding short circuit and rotor eccentric faults with similar fault characteristics. The proposed method takes the harmonic component of stator branch circulations as the characteristic component for fault diagnosis, deduces the harmonic characteristics of stator side circulations when two kinds of faults occur, and summarizes the similarities and differences between them. In view of the weak difference, an LSTM neural network algorithm is introduced to identify it. The possible short-circuit and eccentricity faults of rotor winding are simulated in batches by using the internal fault simulation model to obtain the data set for LSTM network training and testing. Simulation results show that the FFT-LSTM can accurately diagnose rotor winding short-circuit and rotor eccentricity faults of variable speed pumping and storage units.