Power Systems Dynamic State Estimation (PSDSE) using hybrid measurements from Phasor Measurement Units (PMUs) and Remote Terminal Units (RTUs) in the presence of partially characterized and non-stationary Gaussian as well as non-Gaussian measurement noise has been addressed in this work. For Gaussian noise, Adaptive Divided Difference Filter (ADDF) with anomaly detection has been tested in comparison to the adaptive versions of Unscented Kalman Filter (AUKF) and Cubature Kalman Filter (ACKF) for IEEE 30 bus test system and Northern Regional Power Grid (NRPG) 246 bus Indian practical system. In addition, a novel technique termed Adaptive Student's-t Divided Difference Filter (AST-DDF), has been proposed which deals with the presence of outliers in real-time data in the context of synchronous and asynchronous reporting rate of the RTUs. The non-Gaussian measurement noise is assumed to follow student's-t distribution, which intrinsically accommodates outliers. In view of partially characterized time-varying noise statistics, AST-DDF has been validated in IEEE 30 bus and IEEE 118 bus benchmark systems in the presence of bad data and transients caused during the faults in power systems. Results reveal that the adaptive versions of DDF for both the Gaussian and non-Gaussian noise substantiate satisfactory estimation accuracy at a reasonable computational burden and hence recommended for the future Energy Management Systems (EMS).