In the evolving landscape of sixth-generation (6G) wireless networks, unmanned aerial vehicles (UAVs) have emerged as transformative tools for dynamic and adaptive connectivity. However, dynamically adjusting their position to offer favorable communication channels introduces operational challenges in terms of energy consumption, especially when integrating advanced communication technologies like reconfigurable intelligent surfaces (RISs) and full-duplex relays (FDRs). To this end, by recognizing the pivotal role of UAV mobility, the paper introduces an energy-aware trajectory design for UAV-mounted RISs and UAV-mounted FDRs using the decode and forward (DF) protocol, aiming to maximize the network minimum rate and enhance user fairness, while taking into consideration the available on-board energy. Specifically, this work highlights their distinct energy consumption characteristics and their associated integration challenges by developing appropriate energy consumption models for both UAV-mounted RISs and FDRs that capture the intricate relationship between key factors such as weight, and their operational characteristics. Furthermore, a joint time-division multiple access (TDMA) user scheduling-UAV trajectory optimization problem is formulated, considering the power dynamics of both systems, while assuring that the UAV energy is not depleted mid-air. Finally, simulation results underscore the importance of energy considerations in determining the optimal trajectory and scheduling and provide insights into the performance comparison of UAV-mounted RISs and FDRs in UAV-assisted wireless networks.