Reconfigurable antennas (RAs) and movable antennas (MAs) have been recognized as promising technologies to enhance the performance of wireless communication and sensing systems by introducing additional degrees of freedom (DoFs) in tuning antenna radiation and/or placement. This paradigm shift from conventional non-reconfigurable/movable antennas offers tremendous new opportunities for realizing multi-functional, more adaptive, and efficient next-generation wireless networks. In this paper, we provide a comprehensive survey on the fundamentals, architectures, and applications of these two emerging antenna technologies. First, we provide a chronological overview of the parallel historical development of both RA and MA technologies. Next, we review and classify the state-of-the-art hardware architectures for implementing RAs and MAs, followed by a detailed comparison of their distinct mechanisms, performance metrics, and functionalities. Subsequently, we focus on various applications of RAs and MAs in wireless communication systems, analyzing their respective performance advantages and key design considerations such as mode selection, movement optimization, and channel acquisition. We also explore the significant roles of RAs and MAs in advancing wireless sensing and integrated sensing and communication (ISAC). Furthermore, we present numerical performance comparisons to illustrate the distinct characteristics and complementary advantages of RA and MA systems. Finally, we outline key challenges and identify promising future research directions to inspire further innovations in this burgeoning field.