With the rapid development of AI chips, underwater snake robots hold significant promise for navigating complex underwater environments, offering unique advantages in exploration, monitoring, and inspection tasks due to their flexible body and high mobility. However, existing underwater snake robots predominantly employ bulky mechanical configurations with expensive manufacturing costs, resulting in excessive power consumption and limited operational endurance with standard batteries, which impede their widespread adoption and limit their operational flexibility. Moreover, most path following algorithms used in underwater snake robots inadequately account for the dynamic changes in path curvature, leading to serious tracking error in scenarios involving sharp turns or complex terrains, which does not address the demands of more intricate trajectories. To address above issues, this work introduces the U-Snake, a small-sized smart underwater snake robot with a simple lightweight structure and a highly maneuverable controller, adapted to various complicated path following tasks. In particular, each joint of the robot is designed to be small and lightweight to achieve higher spatial utilization, which is covered by a convenient and efficient 3D printed waterproof casing to achieve robust water resistance. In addition, a path following algorithm based on curvature of the path is designed to achieve high performance in various complicated trajectories. Furthermore, an integrated controller combines the algorithm with the kinematics and dynamics models of the robot, enabling precise following of straight and curved paths. The experimental results demonstrate that the proposed control structure effectively guides the robot to follow the desired path.