The non-equilibrium atmospheric-pressure parallel-plate helium dielectric barrier discharge (DBD) driven by a realistic 20 kHz distorted-sinusoidal voltage waveform has been investigated by means of simulations and experiments. A self-consistent one-dimensional fluid modeling code considering the non-local electron energy balance was applied to simulate the helium DBD. The effect of selecting plasma chemistry was investigated by comparing simulations with experiments. The results show that the simulations, which include more excited helium, metastable helium and electron-ion-related reaction channels, can faithfully reproduce the measured discharged temporal current quantitatively. Based on the simulated discharge properties, we have found that there is complicated mode transition of discharges from the long Townsend-like to the "dark current"-like, then to the short primary Townsend-like and the short secondary Townsend-like for the helium DBD that is driven by a realistic distorted-sinusoidal voltage power source. Discharge properties in different periods of discharge are discussed in detail in the paper.