The new generation of artificial intelligence (AI), called AI 2.0, has recently become a research focus. Data-driven AI 2.0 will accelerate the development of smart energy and electric power system (Smart EEPS). In AI 2.0, machine learning (ML) forms a typical representative algorithm category used to achieve predictions and judgments by analyzing and learning from massive amounts of historical and synthetic data to help people make optimal decisions. ML has preliminarily been applied to the Smart Grid (SG) and Energy Internet (EI) fields, which are important Smart EEPS representatives. AI 2.0, especially ML, is undergoing a critical period of rapid development worldwide and will play an essential role in Smart EEPS. In this context, this study, combined with the emerging SG and EI technologies, takes the typical representative of AI 2.0-ML-as the research objective and reviews its research status in the operation, optimization, control, dispatching, and management of SG and EI. The paper focuses on introducing and summarizing the mainstream uses of seven representative ML methods, including reinforcement learning, deep learning, transfer learning, parallel learning, hybrid learning, adversarial learning, and ensemble learning, in the SG and EI fields. In this survey, we begin with an introduction to these seven types of ML methods and then systematically review their applications in Smart EEPS. Finally, we discuss ML development under the big data thinking and offer a prospect for the future development of AI 2.0 and ML in Smart EEPS. We conduct this survey intended to arouse the interest and excitement of experts and scholars in the EEPS industry and to look ahead to efforts that jointly promote the rapid development of AI 2.0 in the Smart EEPS field.