3D multi-object tracking and trajectory prediction are two crucial modules in autonomous driving systems. Generally, the two tasks are handled separately in traditional paradigms and a few methods have started to explore modeling these two tasks in a joint manner recently. However, these approaches suffer from the limitations of single-frame training and inconsistent coordinate representations between tracking and prediction tasks. In this paper, we propose a streaming and unified framework for joint 3D Multi-Object Tracking and trajectory Prediction (StreamMOTP) to address the above challenges. Firstly, we construct the model in a streaming manner and exploit a memory bank to preserve and leverage the long-term latent features for tracked objects more effectively. Secondly, a relative spatio-temporal positional encoding strategy is introduced to bridge the gap of coordinate representations between the two tasks and maintain the pose-invariance for Trajectory prediction. Thirdly, we further improve the quality and consistency of predicted trajectories with a dual-stream predictor. Extensive experiments conducted on the popular nuScenes dataset show the effectiveness and superiority of StreamMOTP, which outperforms previous methods significantly on both tasks. Furthermore, the proposed framework has great potential and advantages in actual applications of autonomous driving.