State of health (SOH) and remaining useful life (RUL) prediction are crucial for battery management systems (BMS). However, accurate SOH and RUL prediction still need to be improved due to the complicated battery aging mechanism. This work combines incremental capacity analysis (ICA) and differential voltage analysis (DVA) based on the second-order RC model with an improved Bidirectional Gated Recurrent Unit (BiGRU) to develop SOH and RUL prediction framework. Firstly, the voltage is reconstructed through the second-order RC model to obtain the incremental capacity (IC) and differential voltage (DV) curves to avoid the influence of measurement noise and the complex parameter adjustment process in the filtering method on the IC and DV curves. Then, a new set of battery aging features are extracted from the reshaped IC and DV curves to improve SOH and RUL prediction accuracy and robustness. Next, the BiGRU method with attention mechanism (BiGRU-AM) is used to build the prediction models for battery aging features, SOH, and RUL. To reduce the impact of the capacity regeneration phenomenon, the Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) method is used to decompose the SOH prediction results, and the decomposed residual is used as the input to improve the prediction accuracy of RUL. The uncertainty of RUL prediction results is analyzed by Monte Carlo (MC) simulation. Finally, the proposed method is verified by experimental battery data from Center for Advanced Life Cycle Engineering (CALCE) and Sandia National Laboratory. Experimental results show that the voltage reconstruction results based on the second-order RC model are applied to ICA and DVA analysis, effectively avoiding the influence of noise. The RMSE of voltage reconstruction is within 0.0006, and the Pearson correlation coefficient between the four aging features extracted from the reconstructed IC/DV curve and SOH is above 0.9. Moreover, this method has good robustness to the cell inconsistency, temperature uncertainty, and a satisfied generalization ability to different battery chemistries, which the maximum RUL predicted AE of CALCE and Sandia battery is within 10 and 5, respectively.