This present study conducts a comprehensive comparison of looped traveling-wave thermoacoustic engines (TWTAEs) and Stirling TWTAEs using two- and three-dimensional (2D and 3D) time-domain models. These models, validated through comparisons with existing experimental results, confirm the accuracy of our developed full-scale numerical TWTAE models. By evaluating the acoustic characteristics and comparing them with conventional standing-wave thermoacoustic engines, this work highlights the superior performance of the Stirling TWTAE. It achieves the highest heat-driven acoustic power and thermoacoustic energy conversion efficiency. This superior performance is attributed to its operation at a lower oscillation frequency (115 Hz) and an optimized phase between acoustic velocity and pressure oscillations (40 degrees). Additionally, our study explores rich nonlinear phenomena within the flow fields of the Stirling TWTAE. Notably, varying the temperature gradients between 410 K and 485 K introduces a bistable zone with distinct pressure amplitudes. Moreover, mass streaming under varying temperature gradients, and mode transitions due to external flow perturbations are observed inside the engine. These findings not only quantitatively confirm the high efficiency potential of the Stirling TWTAE but also demonstrate the effectiveness of CFD in the development of full-scale numerical models for predicting and optimizing the heat-driven acoustic characteristics and nonlinear phenomena of traveling-wave thermoacoustic systems.