Data-driven models have been increasingly employed in smart building energy management due to their superior performance. To avoid performance degradation, the data-driven models should be continuously updated to adapt to the changes of building operation characteristics. During the continual model update process, the knowledge of previous learned task(s) can be lost as information relevant to the current task is incorporated (i.e., catastrophic forgetting issue). Conventional model update methods cannot address concept drift and catastrophic forgetting issues in an efficient and effective way. Continual learning focuses on learning a model from a continuous data stream which can originate from changing input domains or be associated with different tasks. This paper conducted a comparison study on the performance of two advanced continual learning methods (i.e., Elastic weight consolidation, Gradient episodic memory) and three conventional update methods (i.e., accumulative learning, incremental learning, and ensemble learning) for adaptive building energy prediction. The case study on a large open-source dataset demonstrated that, the CV-RMSE of Elastic weight consolidation and Gradient episodic memory decreased by around 14% and 8% on average compared with static model and accumulative learning. The research insights could facilitate the wide deployment of advanced machine learning techniques in the building energy sector.